Requesting entry to the US in a particular visa class when I have more than 1 valid visa

Assuming I currently hold 2 visas for the USA.

• A 10-year validity B-1/B-2, issued a number of years ago but still valid, last used about 6 years ago
• An H-1B visa valid for a couple of months

And I am currently in the US in H-1B status and employed here full-time.

If I leave the US now and travel to one of the adjacent countries, Canada (with a visa) or Mexico (no visa required), and then return after a few weeks (with both of my US visas still valid), can I specifically request to be admitted in B-1/B-2 status instead of H-1B?
I assume that I'd need to convince the immigration officer I'm standing in front of that I won't be working, and that this would be a high bar to reach, but I would have a convincing story...

The goal of this exercise would be to try to extend the final 6-year expiry of my H-1B (since time spent out of the US can be "reclaimed" by application) but still make occasional short trips back into the US (to check up on my property & that the pet-sitter is doing their job) without unnecessarily 'using up' H-1B days.

Similarly to all of the above (and assuming that the answer is at least no worse than "maybe"):
If I hold an H-4 and I'm also eligible for the VWP, can I ask to be admitted under the VWP instead of H-4?
Would there be much point in this, since the H-4 validity is tied to the principal H-1B?

Note: I'm asking here and not on Expats because although I'm currently in H-1B status, it's the possibility of leaving and then being re-admitted under B-1/B-2 I'm asking about.