Procedure to get approved, and transfer balance to a 0% APR credit card?

The specifics of your question are all but impossible to answer. Improving ones credit score is done best by paying your bills on time, but it sounds like you are doing that. A introductory teaser rate is totally company dependent and also dependent upon market conditions, it is impossible to answer this question.

You mentioned that you were worried about "too much credit". With what you listed, your available balances are very low, only $4,300. Even ten times that amount is not very high. I will say your utilization of your credit is high so that may be hurting you.

Keep in mind that 0% transfer offers are rarely that. They typically have a 5% fee or more. So, in your case you would be paying about $125 up front. Even if you were able to pay the balance in full the next month that fee would be paid. Effectively, those zero percent offers work out to a rate of 8% or so if one pays it off by the end of the introductory period. If not paid off the rate typically skyrockets to something very high.

A much better way to tackle this is to find a way to make an extra $1,000 per month for a short period of time. Perhaps this can be done by taking on a second job or overtime at your current job. You might even be able to cut your budget a bit to help out.

Doing that you will, at most, pay about $42 in interest and have about $460 in a savings account. Is 90 days of "pain" worth that? I would think so. Perhaps doing that will help you raise your income.

You might be approved for the additional $2500 without changing anything. Note that if you are, this will likely increase your credit score somewhat, because your utilization percentage will go down:

Current Utilization Percentage = 2500/4300 = 58%
New Utilization Percentage     = 2500/6800 = 37%

Some ways to possibly increase your chances:

1. Call the phone number for the new credit card and speak to a person. Sometimes the CS rep has more information about what will and won't work, and can make recommendations for increasing the chances of it going through. You can also tell them your credit score and they may know if that would be good enough (though, most of the time they have to run your credit regardless to answer that, but it doesn't hurt to ask.)
2. Check to see if Bank B has any offers special offers on your specific card, or with a different card. Oftentimes the bank will have a different card with a good intro offer, and you can move your credit limit to the new card, and even increase it.

I have done both of these with success.

Side note: even if you are approved for less than the full $2500 you can still move some of it to the lower rate. In that case you would work hard to pay off the remaining balance on CC A as fast as possible.