Lighter weight going up the hill, whether the bike or you will make it easier. Lower gearing too will make it easier as you've ask about. But that will also make your climbing speed slower unless you pick up your cadence. But then you'll be bumping up against that self imposed HR limit.
On a 10 speed Tiagra equipped bike, I'd think you'd get the most by changing your front crank to a 3x. That'll require a crank, front DR and front shifter change. Not cheap, but not too expensive either. If your rear DR is a short cage, then you might also have to change it to a mid cage model. But rear DR's in 105, Tiagra or lower tier are also inexpensive.
I just don't think a 2 tooth change on the rear is going to do much for you. And that is probably the limit you should go on the rear without also having to change other stuff that will up your costs closer to what changing the front crank will be.