Step up device or new cartridge?

Another thing to keep in mind is that step-ups can often cause grounding problems and introduce hum into your system, and that's sometimes a devil to get rid of (I've had this problem in the past). Plus, unless you spend a good deal of money for an Expressive Technologies or similar unit, many step-up transformers can tend to add their own distortions and degradations to the signal. And then there's the cost and tangle of additional interconnects, the possibility of impedance mismatches... I think your best bet would be to update the cartridge (I used to have that cartridge, and its good but I think it can be matched by some of the modern higher output designs) unless you can get a different phono card for the BAT with more gain (don't know if this is available).

1. What is the output on your cart?
BAT P3 card is 46db for MM and 60 db for MC. 60 db should be enough for almost anything
Are you sure that your P3 is in MC mode ?

2. Quality of P3 is not up to standalone BAT units. Not even close.
It looks like P10 definitely is out of your budget and P5 has "only" 55db gain in High position. You may try to sell P3 and find standalone phonostage that works with your cart and gives you better quality under $1000

3. I would suggest to call BAT. They are usually very helpful.

You might consider upgrading your cartridge to a higher output Benz-Micro model. I have been using the Benz-Micro H2O straight into a conrad-johnson premier 15 phone stage - very nice.