I will continue offering you that contract until you have no cash left, an event that is certain to eventually happen.

Only if you have an infinite bankroll. Otherwise, there is some tiny but nonzero chance that you lose all your money and the player makes a huge profit. And for the player with the convex utility function, the utility of that outcome is enough to make the whole ensemble of gambles worthwhile.

Then if you extend that to the infinite case by putting the limit outside the expected utility calculation, you will find that the limit is nonnegative too. Or if you don't assume that the result in the infinite case is the limit of finite results, then you have different problems, but then who says the strategy in the infinite case is the same as the limit of finite strategies?

You should name a price less than 0.50 for these options, including a small utility profit for you; I take one option out on each of A and B.

To pick a concave function at random, let U(x£) = log10(x) utilons. And let my bank account contain 10£ at the beginning of the experiment.
U(10£) = EU(9£+A+B) = 1u, so I pay 1£ for options A+B.
Assume WLOG that I'm considering option A first. EU(y+B) = .5*U(y) + .5*U(y+1£). Set that equal to 1u and solve for y: y=9.51249£. Thus I'm indifferent to selling option A for 0.51249£.
After doing so, I am then indifferent to selling option B for 0.48751£.
So I'm back to exactly 10£. No money pump.