Could you inspect the mempool in the days leading up to the prize block and outbid competing transaction's miner fee?

I doubt it, because you state that this is "well-known", so miners would know too. A miner would cherry-pick her own transaction to spend the UTXO. The only way it would be rational for the miner to mine your tx instead of her own, would be if you offered a fee worth more than the UTXO itself.

What's are your tactics to spend it first?

Assuming it's a large enough UTXO to justify it, I'd try to time the purchase of some cloud mining, and try extra hard to win that block, and include my own tx in the block.

Curious to hear the answer about propagating a "not-yet-but-soon-will-be"-transaction. I believe it would, or at least should, propagate.

Fun discussion.