So then Picard could refuse the help, bring up the shields to max, and cut himself out of the asteroid with phasers. Even with the danger of a collapse, it's not really a danger as I don't see a bunch of rocks causing any major issues with the shields up. Then to the Romulans, you just cut your way out of an asteroid, no phase cloak needed.
Worf suggests that, & (Per the usual) gets shot down, when Data says the asteroid's internal structure is unstable. Cutting into the blocked entrance with phasers could likely cause the entire chasm to collapse, crippling & burying the ship deep inside an asteroid about half the size of a small moon
Then just turn on the phase cloak. (I should have thought of this one years ago!)
If you can't fire while phased (which seems reasonable), then activate the cloak a split-second after firing torpedoes. If that's not enough time, then put the torpedoes on a delay. Make sure you blast your way out on the side of the asteroid opposite the position of the Romulan ship so they can't see you emerge. Then just sail out (whether the opening is actually large enough for you or not). The instant you emerge, turn off the cloak. Then contact the Romulans. "Oh, it seems we don't require your assistance after all. Yes, that is quite a small opening. We barely scraped through thanks to the skills of a gifted pilot. Anyway, have a nice day!
"Helm, warp three. Engage. And hurry."
There are probably several holes in this idea, but the fact is Picard wouldn't have done it anyway because he would have felt honor-bound to declare the situation to the Romulans.