It would have been stupid to turn Hart heel after investing so much in him as a fan fav during the previous two years, especially if this was a "pass the torch" moment for Brett, establishing him as the true No. 1 guy, not just the prerequisite good guy champ keeping the belt warm for Hogan's return. Vince wanted Brett to have that credibility with the audience, something he lacked as long as Hogan was in the picture.
It is true that some of Hogan's appeal had gotten stale, but as someone who was watching wrestling for a decade before Brett Hart ever won the World Title I can tell you in WWE no one was more popular than Hogan, at his height Ultimate Warrior was close, but until Steve Austin, no one sold tickets and drew crowd reactions like Hogan. If the match took place as was rumored with both being faces I think Brett would have gotten a decent pop but Hogan would have been the crowd fav. Hart was never close to eliciting the kind of fervor from crowds that UW did at his peak, and he could only split the crowd 50/50.
I do think if Hogan would have put Hart over clean it would have been a huge rub for Hart, his lack of charisma though was always going to hold him back as guys like HBK & SCSA ascended up the ladder. Of course WWE didnt have a good undercard back then (93-95) and Hart wasnt the only main eventer that was hurt by lousy booking (as bad as Hart vs Isaac Yankem, evil dentist was, Mabel as a World Title Contender vs Kevin Nash was worse).
While Hogan may have been losing some steam, it was interest in wrestling in general that was on decline in early 90s. Outside of Memphis the territories and independents were dead, once Jim Crockett Jr divested from NWA/WCW they went down the proverbial toilet, and WWE got very formulamatic & dull, the audience went away slowly. Much of that audience paid cursory attention to WWE but were more interested in other feds, they quit watching as their promotions declined. A lot of people blame Hart for WWE's decline. If you look though the WWE audience was already in decline before he became the program centerpiece. I dont think he had much charisma, I was always way more entertained by contemporaries like HBK & Nash, but was a good solid hero character, kids loved him, and he could deliver a good match at any big show (basically John Cena today, Hart was better in the ring but Cena is way better on the mic). Now Hogan I dont remember getting too many mixed reactions during this time. I remember Hogan being way over vs Taker. Against Flair it was really an NWA thing, there was always going to be a significant portion of the audience that supported Flair over Hogan because they were NWA fans and saw him as a better entertainer. Still, although this was present somewhat in their WWE bouts my memory was that the WWE crowds were mostly for Hogan. I saw them wrestle twice in Pgh at WWE shows and Id say the crowds were 70-30 Hogan. Hulk didnt really hit the wall in terms of being stale until he went to WCW, although part of his problem then was that this audience was comprised mostly of fans who favored their guy (Flair) making it hard for him to generate the kind of positive crowd reactions he was used too even when their ratings were good, pretty much the opposite dynamic that defined their bouts in WWE were Hogan had the "hometown advantage". The only time Hogan really got booed in WWE was initially at The Royal Rumble when he interferred to cost Sid the match and the vacant title. That was lousy booking by WWE, thinking just by making Hogan mad you could turn someone heel with the audience but the whole script really made Hogan look like a jerk and made Sid look sympathetic instead (an every man for himself battle royal, promoted for the possibility of seeing allies go against one another, where Hogan had already gone against allies like Piper & Savage, Sid is left in the ring with an exhausted Flair, who has wrestled for an hr at this point and a fresh Hogan, so Sid dumps Hogan first to take his chance with Flair, and as Hogan looks on from ringside Sid tells him "every man for himself", which was true. Hogan's very un face like reaction was to argue with Sid, grab him, and distract him long enough for Flair to dump him out for the win. If WWE thought that would make the crowd turn on Sid they were wrong. They made Hogan look bad and the audience booed Hogan instead.)
Eventually they did enough to make Sid look bad and crowds cheered Hogan. Their WrestleMania Match didnt generate as much heat as Flair-Savage or even Hart-Piper because the build up and hype was weak in comparison, not because Hogan wasnt popular anymore. During the match the audience is clearly behind Hogan.
In the end, would it have made sense for Hart to turn Heel, given the huge investment WWE had made in his Hitman persona, with heels like Yokozuna & HBK already around, absolutely not, at least not if their goal was for Hulk to put Hart over as a legite heir apparent as the No. 1 guy. In a face vs face match between them I believe Hogan would have been more popular initially but the audience would have accepted Hart if he had won. Everything else being the same, I dont think it would have changed much for either moving forward.