I hadn't thought of her striving to win the game in that scene as a metaphor for her attempt to dissuade Edmund from the clergy, but it does makes sense. I have thought that she sees relationships as a power struggle so I agree with the competitive element.

I wanted to add, I've always thought that "No cold prudence for me. I am not born to sit still and do nothing" was a direct dig at Edmund.

It directly relates to what she said in ch. 11
“Oh! no doubt he is very sincere in preferring an income ready made, to the trouble of working for one; and has the best intentions of doing nothing all the rest of his days but eat, drink, and grow fat. It is indolence, Mr. Bertram, indeed. Indolence and love of ease; a want of all laudable ambition, of taste for good company, or of inclination to take the trouble of being agreeable, which make men clergymen. A clergyman has nothing to do but be slovenly and selfish...

One of the things I've always thought was telling about her character in ch.25 is this
was considering Sir Thomas, with decided ill–will, as the destroyer of all this, and suffering the more from that involuntary forbearance which his character and manner commanded, and from not daring to relieve herself by a single attempt at throwing ridicule on his cause.