Sacrifice The Rigid Conversion Factor
Your question makes it sound like you have a strict 60:1 conversion factor, because you mention a minute expanding to an hour; your response in a comment makes it sounds like a 10:1 conversion factor, because of the six-second round to one minute conversion. But in either case, you seem to have some constant or rigid conversion factor in mind.
So what you're asking for is a solution to of the classic problems of split parties-- keep both groups occupied (1) with appropriate shares of the limelight (2) for the same amount of real-world clock time-- but now with the added complication of (3) except the in-game clocks are running at different but known (at least to you) fixed rates.
Hopefully when I lay it out like that, you'll see why that's basically unworkable. Combats, generally, don't have pre-determined lengths, and every round that accrues in the main-party is accruing ten (or sixty?) in the bubble. And if your character is doing something with turn-based mechanics inside the bubble, it's probably hard to predict that accurately as well.
Give Your Bubbled Character Something Open-Ended To Do
I have no idea what this 'something' would be. It has to be appropriate to your game, and the area they're in. But it needs to have the following characteristics:
It needs to be somewhat open-ended and light- or non-mechanical. You need to not be counting rounds or turns while doing this
It needs to be something you as a GM can draw out or contract so that when your main-party combat is over, you can plausibly and satisfyingly end the in-bubble segment
It needs to be something that can plausibly take approximately (and I cannot stress that enough) the amount of time you need to fill, here. So if you expect your main-party conflict to take a minute, and your conversion factor is 60:1, pick something that could plausibly take somewhere between half an hour and two hours.
It needs to follow the other general guidelines of good role-playing situations-- the character has to have something to do, choices to make, information to receive, etc.
So things like needing to find something, talk to someone, build something, catch something, etc are all reasonable things.
Magical Hand-Wavey Narrative Sync-up
The end goal, as is probably obvious by now, is something like this:
- Bubble character wanders off into the bubble
- Main group starts combat
- After each round (or maybe half-round, or whatever seems right) of combat with the main group, break off and give your bubble character some interaction, and probably some early heavy indication of what they're "supposed" to be doing
- Switch back and forth between main group and bubble character
- Use your existing GM skills to telescope the bubble character's task so it ends around the same time as the combat
- Aggressively do not care about an exact conversion factor, just sync the characters back up as appropriate.
The good part of this scheme is that you're not doing anything much beyond the ordinary skill of a pretty skilled GM-- you're not doing something superhuman like trying to anticipate rigid time in two domains.
The possible bad part (although I don't see it as one myself) is that if this sort of thing is a feature of your world, it almost automatically imparts a sort of a fey, unpredictable, chaotic air to the phenomenon-- maybe one time the conversion is roughly a minute to an hour, maybe next time it's a minute to a day. So if your characters are the type who are going to want to rig up stop watches or try to figure out and exploit an exact conversion factor, that can cause issues.
(I myself would lean into this hard, and describe the course of the Sun as moving from north to south as time passes, just to underscore the unnatural weirdness of it. But that's just me.)
Disclaimer
I am uncertain whether to cite this as an experience-based answer. On the one hand, I have extensive experience playing in a game (Amber-inspired) where the default condition is for characters to be running in different worlds and at different speeds. And this technique of not sweating the details and just syncing up the characters whenever they meet is how it is handled. (Who knows what Dworkin was thinking when he made the universe, it just works this way. The hardest thing about time is doing it.) It fits the genre we are trying to emulate very well, and I wish I could take credit for formalizing it that way, but it wasn't my idea.
On the other hand, we have not to my memory ever tried to use those techniques to solve this problem. Amber in general, and this game in specific, does not even pretend to model things at the level of seconds, minutes, or combat rounds.
But certainly the core idea-- do not use fixed conversion factors-- stems from that experience.