Depends on what you want to do and your resources. Maximizing what you do, you could fly back to Thailand, get your ID and new passport (or do the latter in Jakarta) and return to Jakarta for the rest of your stay then come back to Thailand. The cost of this is a round trip flight Jakarta-Bangkok and accommodations in Bangkok.

As others are saying, there are other alternatives: return now to Thailand and settle things in Bangkok (loses out on rest of Indonesia), stay in Indonesia return on a visa exempt stamp and have to get a new DTV, negotiate a way to get a new ID sent to Jakarta (you are not certain it will work) and handle things before you return, enjoy Jakarta but return to the UK to sort things out before returning to Thailand.

None of these are ideal. You have to assess their probability of success, costs (financial and otherwise), and how they work, and decide which way forward you want to go.

I don't see your description as a Catch 22 situation. It is a situation where you may have to have duplicative coverage in the short term, but there are cheap health policies that meet the requirements and give virtually no coverage. Some of us did this during the pandemic - bought Thai policies because our own policies did not provide adequate certification. (I would also check coverage for long term international stays and coverage for care/incidents in Thailand.)