Well, he does have to sign anything the House may actually do for it to become law, yes?
You can’t unilaterally accomplish much if you only control the House. Beyond your own investigations/oversight.
Sure, you can, and should, send up lots of good/decent legislation that will go nowhere in the Senate. Show you are trying and Trumpublicans are obstructing good things getting done.
Do both
Sure. But don’t beg for Trump’s help. Make it politically impossible for him to keep this crap up.