My unit also failed and I investigated it. This is a model 100 for remote control. (It is not for streaming as discussed by Gekko above.)
It is correct that the label is held on by heat sensitive glue. Warming it up will allow you to peal it off without losing the serial number tag. The two parts are NOT held together by screws but tabs/slots both top and bottom since there is some flexibility with the plastic as they snap together. Unfortunately at both the exposed and unexposed ends they are fouled with glue and can not be disassembled without damage.
Once the unit is opened, the battery is easily replaced it you can get one. It is Lithium-Battery CF234044, 3V-/0.6Ah as shown above with a connector. Easy to change. The hard part is to get the replacement. Searching yields four places on the web – all of which were on Ebay. One in the states where it was about $25 and the others resolving to one in Germany (Lithium-Batterie CF234044 online kaufen | Pollin.de) for €0.35. Shipping in Europe is reasonable but to the USA was outrageous (€72.) unless you can convince them to ship as a small packet. This may change, of course.
Actually my unit was flaky which is not how a battery dies and I saw it was the power switch. The operating toggle fell out and was lost. Also, I would have had to tape or glue the unit closed after replacing the battery. Since I was due another unit (with a warranty) as I recently got a new set of hearing aids, I dropped the project.
My personal experience with StarKey is that they replace rather than repair units returned under warrantee. Out of warrantee they give a credit for the broken unit which is effectively a price reduction.
I note that there are units for sale on the web for much less. Of course, you need to be able to get your hearing aids re-set to the new remote id to work.