Based on the request for a hard-science answer instead of the one I put above. I'm keeping the other one because it is a lot more accessible. Exploring surviving the heat death of the universe calls for exploring the fringes of math, society, science, and survival.
In comments, you mention "I guess the question I'm getting at is if there is some impossibility theorem that says "all bubbles during the heat death pop" or something. " The answer is no, there is no impossibility theorem like that, but you asked for the wrong theorem.
The first issue is defining what a bubble is. If your bubble is viewed as "part of the universe," then heat death has not yet occurred because your bubble contains information. So you clearly intent to treat the bubble as "outside of the universe," which by the same definitions as heat death means you cannot make classical measurements about the state outside the bubble. See Maxwell's daemon for an argument why.
Now the problem becomes that the universe is not reborn in $10^{10^{56}}$ years. Quantum mechanics doesn't do things on a schedule. What that theory actually states is that it is possible that, through quantum mechanics, a waveform similar to that of a universe (such as a dirac delta function that could serve as a big bang) will occur due to random effects. Two key takeaways here. First it assumes that QM is 100% correct that there is absolutely no correlation between particles which cannot be modeled as random variables, and that those variables all are subject to the Central Limit Theorem. In this case, the rebirth of a new universe will occur at a time also specified by a random variable with an expectation of it occurring at $10^{10^{56}}$ years. If you popped your bubble too early by 0.000000000000000000000000000000000000000000001%, you would find that you would be so far off that even a "bubbled" black hole with all the known mass in the universe would evaporate before genesis reoccurs. If you pop it 0.000000000000000000000000000000000000000000001% too late, you will miss genesis of the new universe, miss it evolve, and wake up during its heat death after all of its black holes have vanished. In fact, you wouldn't even be able to tell the difference between the two cases, because there would be so little information in the universe to study! (and actually, the bounds are much tighter than that. The actual bounds have roughly $10^54$ zeroes in them! I just didn't feel like holding the zero key down so long that we would start to enter heat death of the universe before we finish this answer!)
This means we need to relax the "bubble" requirement. We need to make measurements of the space outside to observe the new universe happening. One approach would be to rely on purely quantum interactions which transfer no energy, but the best that's going to do is give us a probability of the bubble popping at any given time. We need to make real measurements. This means energy goes out of the bubble, and energy comes into the bubble. Now we are part of the universe, because we are interacting thermodynamically.
One of the side effects of this is that we now apply thermodynamic laws, which state that energy will always transfer from the hot side to the cold side. The universe outside is the cold side, so energy will seep out. You are now playing a game of Russian Roulette: will a random universe appear outside before the same forces that make random universes tear apart your bubble.
As I mentioned in my previous answer, there is a solution to this. Consider survival not as a pass/fail thing, but a metric. Zero survival is "no information stored in the bubble when it was sealed survives," maximum survival is "all information stored in the bubble when it was sealed survives," and the metric in between captures how much of the important information you keep. This forces you to understand what you actually want to keep and what you are willing to lose. Over time, you sacrifice the lower value information to provide the energy needed to do measurements of the universe.
If you use an amount of power proportional to the energy you have stored, the pattern works. At first you lose more energy than you gain information, because nothing interesting has happened. As you continue, you expend less and less energy, exponentially. Such a curve only ends when you run out of bits of entropy. As long as you expend a small enough amount of power, you can let the heat death of the universe outrace you, and wait for the new begining.
The limit of this is when you run out of bits of entropy. When you only have 1 bit of entropy left, you cannot subdivide it. Or can you? What if we reframe the metric. What if the metric permitted fractional bits of entropy. A fractional bit of entropy could not cause a bit flip from 0 to 1. However, if the universe that is reborn also has some constructs which can be described as having fractional bits of energy. We might be able to influence the new universe, even if we can't provably flip a bit from 0 to 1.
How could this work? We went to two extremes. Quantum interactions provided no classical information to decide when to pop the bubble. Classical measurements provided that information, but had entropic costs. What about in between?
There is some recent development in Weak Measurement, which is a QM approach to gathering information about a system while learning much less about the average state of the system than usual. For example, consider a case of polarization. We are used to measuring photons classically to determine horizontal or vertical polarization. We are used to measuring photons with pure quantum entanglement, but that doesn't generate any information. Weak measurement provides results like "There is a 10 degree difference in polarization angle between the measured photon and the photon you just weakly entangled with it." You still don't know much about the angle, but you know a way to calculate it.
This approach can permit measurements which have the appearance of a fractional bit of entropy associated with them. This is demonstrated for such QM operators using the "gentle measurement lemma."
This suggests that, through weak measurement of the dying universe around you, you could interact with the world, peering out for a new universe. Once you see that new universe, you could influence it using weak measurements to make sure every fractional bit of entropy you saved gets its maximum impact.
And, if you got lucky, and the new universe spawned early, you might even make the choice to classically measure whether the new universe has formed, at the expense of energy. If successful, you could even interact classically with that universe.
Construction of this is the tricky part. In particular, it is impossible to prove that it will work using modern mathematics. To ensure the power used is proportional to the energy in the bubble, your bubble making engine needs to "know" how much energy there is. However, to do that, it needs to know how much energy is in the energy knowing system, and so forth. As it turns out, modern set theory has an axiom, the Axiom of Regularity, which forbids such self-referential structures. You would have to use one of the alternative set theories that are being developed which permit urelements such as Quine atoms, and then you'd need to go make sure your scientific theories do not fall apart in such systems (such set theories are not "well founded" which has some pesky consequences).
In the end, you may be able to do it, but using existing math and science it would be impossible to prove that the bubble you constructed will do the job, with 100% success. You'll always have some uncertainty.
But, perhaps, when dealing with the certainty of heat death, maybe we can accept not knowing the future in exchange for avoiding a known undesirable fate.
