How to independently verify that bedbug heat treatment is working?
First, I'm not a lawyer. Second, I know 100% that a landlord is required to provide habitable living spaces if they are renting for occupancy purposes. A proven existing condition such as a bedbug infestation is near certain proof that he has not (yet) provided habitable living accommodations and there is already precedent set when the rental is not habitable; therefore it would be advisable to seek legal counsel to ensure you are paying only if you should be. Further, if I recall correctly, for a situation like this the landlord only needs to prove they are taking the correct actions to resolve the issue. E.g., using extermination services is such an action. Now, it is questionable whether your business relationship is off to the right start, you be the judge,  and let legal counsel assist. 

The real answer to your question is that you cannot be reasonably certain. There simply is no way.

Also, if everything you've said above is on the table and nothing is left out or massaged, then technically it is not your job, nor should you really want it to be your job to solve the problem. Unfortunately, bedbugs are hitchhiker bugs and they can get anywhere, any number of ways. The landlord could have successfully exterminated the issue in the apartment, and one of your belongings could have brought it in, even if you didn't have any issue where it came from. Far fetched? Sure. But still possible.

Numerous articles and research identify heat treatment as a sure thing, presuming the heat gets to the bugs, and it has to be at the right temperature for the right length of time. A series of treatments with different chemicals may be employed, dust in outlets and switches, with the covers off. And now that you have possessions on the premises, these should be treated as well. Also, most exterminators will recommend a re-inspection and re-treatment around 20 days after the initial treatment to get any that have since then hatched, and a second shot at what could have been missed the first time.  If there was truly "an easy" way that was not obtusely cost prohibitive to detect success, exterminators would employ it, or be required to employ it by the paying clients.

I have experienced the unfortunate gift these pests brought me by finding their way into a duplex. Of course while it had been occupied by the same people for a very long time, the fingers were pointed in all directions. I chose the path of whole house as to avoid misses and while clearly I didn't introduce the problem, I was going to be stuck with it no matter what. It was extremely costly, and the job was not performed well at all. On the second treatment I had the owner of the company involved and they had two people perform the work and took their time. Like I said, it was not cheap, but the problem is thankfully gone.

Your ideas are all fine, but they are not fool proof and there aren't any foolproof options, so any suggestions you get are only that - suggestions. At the end of the day the only SURE PROOF you get is TIME, the longer they have no food the more die off, but even then I have read of people seeing them months afterwards *(no idea how, but still, you read about it).