What ruleset?
Under Unified Rules, I'd probably favor Crawford. The only "advantages" Punk has are a greater -- albeit still rudimentary -- knowledge of kicking and BJJ. Low kicks are the bane of a boxer and Crawford is probably fairly ignorant of them beyond what he's seen on TV. He seems to have a little bit more familiarity with submission grappling based on some playful tussles of his caught on film:
Personally I think if Phil tries to chop Terrence down he's probably going to get clipped with something over the top before he wins that race more often than not. He doesn't strike me as the kind of guy who would be able to cleverly set up low kicks behind hands or level change feints in a disciplined manner, after all... but he does strike me as the type to telegraph his intentions to kick low, drop his guard, and eat a big overhand in a very poor trade. Also Crawford is a switch-hitter by nature so even if Punk does manage to compromise his lead leg, Bud could potentially switch stances and still win the fight. Low kicks are one of Punk's better options, but they're not a silver bullet by any means.
It might be a better idea for Punk to try and exploit his size advantage by closing distance, making the fight dirty with a lot of clinch work against the cage, dragging Bud down, and hunting a finish with a submission or GnP. But considering the massive gap in athleticism and the fact that Crawford is a competent wrestler, doing so might involve some kind of unorthodox guard-pull or something else really desperate by Brooks. Punk looked fairly helpless on the mat in his UFC bouts against journeymen. The fact that post-UFC he competed as a Blue Belt Senior at a local, small-time BJJ tournament and still came in last place against middle-aged, amateur-level competitors (arguably glorified hobbyists) doesn't really inspire confidence. He lost one match 2-11 (he had a moment or two of success where he was able to get on top and attempt a kimura) and got dominated in the other 0-7. In his defense, that was in the gi and the guys he was competing against currently have decent records on the Chicago local grappling scene.
It's also worth noting that Crawford grew up wrestling in Nebraska (a legit wrestling state) and has continued to dabble in wrestling throughout his career to this day. He got his kids involved in wrestling and frequently attends their matches and coaches them. He oftentimes fools around with experienced grapplers for the hell of it or as part of his training:
That's Bud embarrassing and pinning a member of Kazakhstan's national wrestling team in a playful match while wearing blue jeans and flip-flops. And Kazakhstan is by no means a backwater of grappling talent; they have a rich culture of folk wrestling, they've claimed 18+ Olympic medals (including gold), and they've also medaled at U23 Worlds.
Bud is also an extremely skillful in-fighter and clinch technician by boxing standards and you can see him use these skills in the ring frequently. He's got sturdy frames and solid posts. He immediately pummels for underhooks and he's clearly super strong & robust for his size and thus difficult to move around as a result. In fact, he has been known to be able to tie up and push around ostensibly bigger, stronger boxers like Canelo.
Take a look at this. I know Bud's a bit bigger than his "opponent", but he's toying with him -- playful hand-fighting, laughing and talking to the spectators, not even attempting offense until the very end. The kid looks like he's going all-out trying to take Bud down or at least gain a dominant position on him and he's getting manhandled.
I'm not convinced that Punk would be able to reliably take Bud down or hold him against the cage. Here's his fight against Jackson, who Punk makes look very good...
Brooks attempts nearly ten takedowns in this fight and only succeeds on one, and it's atrocious from a technical perspective: he finishes on his knees, scooting along the canvas. I think the only reason he got away with it is because Jackson was off-balance from entering the pocket and throwing a big shot at the time Punk made his entry. And despite all that, Phil only ended up with a few seconds of top control before Mike casually stood back up from under him. Whenever Punk tried to shoot on Jackson or hold him against the cage, Mike casually spun him around and reversed the position. Then there's Punk's striking in this fight... yikes. I'll give him credit for actually showing some semblance of a jab and straight, but other than that it's just a lot of head-hunting & over-swinging with telegraphed haymakers. Also lots of wild flurries of punches in an attempt to crash into the clinch... where he inevitably gets manhandled. Punk only throws a handful of kicks in the fight and they're all fairly non-committal and most are totally off-target. Jackson visibly stuns him multiple times with some really basic fundamental boxing, so Crawford in 4-ounce gloves might kill him.
EDIT: I'm only just now seeing that you gave them both six months of prep. That just makes me more confident in Bud. Claressa Shields had about that same amount of time to train for her PFL debut, which she won by TKO -- despite not having Bud's wrestling background or punching power to fall back on. Not to mention that unlike CM Punk, Claressa's debut opponent -- while a bit of a can -- was nonetheless a BJJ Brown Belt with 12 ammy & pro MMA bouts and lots of competitive grappling experience.
Meanwhile, I sincerely believe that CM Punk wouldn't get any noticeably better with the six months provided. Roufusport poured all they had into him for over a year straight before he made his UFC debut, where he got dominated. He had nearly two years to train and improve before fighting a can and getting dominated again. Left MMA, kept training in BJJ and tried his hand at that... same result. Nothing against the guy, but he just came too late to combat sports and lacks the aptitude for it I guess.
Bud is the opposite. He is already a multi-disciplinary combat sports athlete who is a generational talent in one art and competent in another. With that foundation to build upon along with his advantage in youth & raw athleticism he's way ahead of the game. If he dedicated himself fully to the task and trained at a decent MMA gym he would pick up what he little he needs to: getting comfortable with the small gloves, fundamental defensive grappling, and preparing him to deal with low kicks. Even if Punk spent six months working on a specialized "boxer-killer" gameplan centered around low kicks and submission-hunting, I still think Crawford finds the TKO... or maybe he just hurts CM Punk multiple times en route to a clear Decision victory. His knockout power noticeably dropped the moment he started fighting larger guys, though those were other elite boxers.
One final bone to throw to Punk: it's worth noting that he would likely have a legitimate size advantage. He's 6'2" with a 73" reach and had a billed weight of like 218 pounds in the WWE. He almost fought as a UFC Middleweight instead of Welterweight and stated that the last time he was 170 pounds was in middle school. Meanwhile Bud is only 5'9", though his reach is actually slightly greater than Punk's at 75". Bud from the time period dictated in the OP was still boxing at 147 pounds, basically a UFC Featherweight. He didn't begin bulking to move up and face the likes of Madrimov and Canelo at 154 & 168 pounds until like six years later. That being said... six months should be enough time for him to put on a bit of mass in between his other training endeavors. Besides, we've already established that Punk isn't really an elite grappler capable of exploiting a size advantage in the first place and Bud has shown the ability to hold his own in the clinch against bigger, stronger opponents. Also, I can't harp on this enough... I expect the speeed & athleticism gap to be massive and more than make up for the difference in size.
