My 125A Square D HomeLine will not accept tandem breakers
If you need the circuit count, you need the circuit count. That is something you will need to decide.
if money is an object, you can install something like a 200 amp MLO (main lug only) panel and backfeed the panel with whatever size breaker your current service is. Make sure the panel is rated as service equipment. That way, you would not be changing anything on the city side of the main breaker.
You can even use a 200 amp MB (main breaker) panel and just not use the main breaker but backfeed with the smaller breaker. Just be sure, in either case, that you put the "service disconnect" sticker by the appropriate breaker.
It would also allow you to eventually upsize to a 200 amp service in the future using the panel you already have.
Even less expensive (generally) is installing a sub-panel.