The "solution" to this is to agree to gate-check the bag, but to request your right to declare "excess valuation" of the contents to increase the level of carrier liability (and to pay the relevant fees mandated in the Contract of Carriage for the excess valuation). The airline is obliged to permit you to do this at the point of luggage check-in.

In every single case I've requested to do this, the agents back off pretty much immediately. It's easier to let you carry the bag on board than to deal with the paperwork for the excess valuation, which would almost certainly either delay the flight or wind up with the flight departing without you.

Note that this only works if you are within your baggage allowance and passage is being denied purely because of "space on board" and not due to non-compliance on your part. If your bag is too large or too heavy, I would not suggest you try this.