Will I win this game with Capital One?

I have a stupid card with them with a limit that is less than a grand. It is pathetic. I’ve asked them to raise it several times and they said I cannot initiate the process; they have to do it from their side. I haven’t got a limit raise in the last two years.

I don’t want to close the account because it is the oldest and I don’t want the credit score hit. If I call them and tell them I’m sick of their low-ball scheme and want to close my account will they counter by offering to raise the limit? Does anyone know if I open a different card with them if I could then consolidate this account into the new one without taking the credit score hit?

I am sick of this lemon.

Please do not lecture me on credit card responsibility.