tempest, I tacked ahead of them and under them, they did not need to change course and they did not change course.
Then it appears that you are claiming that Rule 20 would apply here " Calling for room at an obstruction" ?
The leeward boat ( you) calls to the windward boat for room to tack at an obstuction. The windward boat can then either tack and give you room, or, if they feel that you have room to tack under them can call back and say
" You tack" If the windward boat tacks to give you room, then you must tack immediately after, you cannot continue sailing, even if it might give you advantage.
In your situation, it sounds like you did not have room to tack and the windward boat needed to give you room to tack or, tack away.
The other boat seems to be claiming that though you were leeward in the narrow channel, you placed yourself in the situation by making two choices,
One, to tack under them rather than dip. And, two, by not leaving yourself enough room when doing so to tack away from the obstruction without forcing them to tack.
So I would ask, by what rule does the other boat claim that they were not required to honor your request for room? Leaving aside for a moment the fact that they interfered with your boat by pushing it