2B. If Segwit has not activated at height H, Input 1 of the Bounty Payout is not valid since it spends a P2WPKH output, preventing the miner from including the Bounty Payout transaction in the block. (However, the output of the Segwit Assertion tx can be claimed since it is treated as anyone-can-spend, although this is not an issue since it is a very small amount).

It's a small amount by itself, but miners who are aware of Bounty Payout Transaction will try to include both these transactions (and both are valid both on SW and non-SW chains by definition of SW being a soft fork).

If you set timelock of BPT to (H+1) then you sort of discourage this behavior because a miner of block H might be not the same as miner of block (H+1), thus he cannot grab this bounty for sure.

Still, there is a chance that same miner will mine both blocks, so game-theoretically it makes sense to insert SAT into your block since your expected payoff is positive.

So I'm afraid miners will just grab these bounties regardless of segwit activation.