USAA, Pentagon & Navy Federal Credit Union frequently have promos where there is no balance transfer fees and low/0% APR financing if you qualify.
And OP, they would have to run a credit check in order to tell you what limit they can give you. And to run a credit check, you would have to agree legally to "apply" for the card. So I don't think you can do that.
Your best bet is to ask the company what their approval criteria is for your target credit limit. They may say, credit score of XXX, less than 30% debt to income ratio, job stability 2 years or more, etc. etc.
Good luck!