Yep, the TLV6703 looks near ideal for the application. It ticks all the boxes except novice friendly through-hole, and enough output current capability to directly switch significant loads.
However, as T.I. say it can be used on a 'DIP Adapter Evaluation Module', putting it on a SOT23-6 breakout is unlikely to cause instability, so the rest of the project can use through-hole construction. If necessary, the limited output current capability is easy to solve with a PNP emitter follower, permitting beepers with >40mA inrush current to be used.
Unfortunately there's still the issue of getting the average beeper current low enough to be a negligible drain on the battery, in case the alarm isn't noticed for a while. Ideally you want the average current to be smaller than the expected self-discharge, which for a typical smaller automotive battery can be as low as a few mA equivalent discharge current. Therefore the beeper needs to be efficient, and also pulsed with a low duty cycle.