Normally I'd like to apply my mind to this kind of dilemma, but unfortunately someone has beat me to it and done a much better job.
Firstly in terms of your credit rating, there are a number of websites that will offer it to you for free. I personally use Noddle, but I believe CallCredit do it as well. (Side note: avoid the ads and do your own research rather that what they may offer you, they are still a business). This may be a good start to give you an idea of the kind of offers you may get from credit card companies.
In terms of what you may be offered, Moneysavingexpert has an "eligibility calculator" which should give an impression of how likely you are to be accepted for a particular card. (I think Moneysupermarket offer a similar service, but then they are the same company now anyway). This may give you least chance of unnecessarily damaging your credit rating.
On to my favourite bit (the maths). Based on the numbers you listed, it sounds like you are currently paying around 14% rate of interest. If you do a balance transfer you are likely to pay around 3% fee (possibly lower, but best to overestimate). That means on £1900, you will pay £57. Your minimum payment is usually around 2.5% of the outstanding balance (Approx the £45 you stated). You shouldn't drop this each month because it will take even longer to pay it off. If you pay £45 every month, then it will take about 3 months to make the balance transfer worthwhile.
The advantages (of the balance transfer) after:
12 months balance transfer: Approx £190 better off
18 months balance transfer: Approx £300 better off
The market leader is 40 months, but that will only be available for a select few I'm guessing.
You would need 44 months before the balance transfer would be paid off. You would need 59 months to pay off without the balance transfer.
I would say if you can afford it and you choose a balance transfer, divide your balance (after the fee is applied) by the number of months the balance transfer is for (If you can afford more than this calculation, do that instead).
This way you can be certain not to pay any interest on this again. This does also give you the flexibility to reduce this if necessary in harder months... BUT you would then pay interest at the end of the interest free period.
If you need anything more or anything explained further, I'm happy to help. :-)