I've found some alternative options. Unfortunately, though, it seems like most of them won't work most of the time.
In-kind transfer to another custodian
This is where your investments are just transfered to another custodian without being sold, so they don't change at all. This is the best possible outcome IMO, because you'd be able to choose a reputable one, and you'd have control over it.
Many custodians won't do them, though. In some cases it might depend on whether or not the new custodian provides the same investments, but some custodians won't do it even if they match.
Dollar-cost-average the potential losses
You can liquidate a portion of the investments yourself, and transfer them to a new provider. If you do that 4-5 times, it would split the volatility across multiple time periods, to average out the fluctuations.
Some custodians don't allow partial transfers, though. Even if they do, some custodians only transfer the money via physically mailing a check to the new custodian, and then entire process can take 1-4 weeks. Those delays can limit the number of times you'll be able to do partial transfers, decreasing the effectiveness. That might also greatly increase the time window where your money isn't invested, increasing the risk of the purchase price being far away from the sale price.
Your HR team may have enough leverage with the custodian to convince them to make an exception for your company, and do an in-kind transfer, or reimburse you for any losses they cause, etc.
Leave your HSA with the current custodian
In my case, WageWorks was my benefits provider, and was being acquired by HealthEquity, but they partnered with BNY Mellon to be the custodian of HSAs. One WageWorks representative told me that some employers were allowing people to leave their HSAs with BNY, but BNY told me that HealthEquity wouldn't let them do that. I'm not sure who was correct, but it may be possible in some situations.
Deny consent to liquidate
Another option that a benefit provider suggested was denying consent to liquidate the investments. I'm guessing this would work similar to above, but forcing it to happen through legal means. Our HR team is looking into it, and I'll update the post when I have some definitive info.
Just let it happen
Like Ben said, doing the liquidating/transfer as fast as possible helps mitigate the risk. If the acquiring custodian can do it faster than the you could transfer to a new custodian yourself, then just letting them do it may be your least-bad option.