Once the funds are in your checking account, technically (and actually) you can do whatever you want with no issues. The only time you'll have a problem is if you have a cash access check and you make it payable to the issuer it came from, which isn't allowed... otherwise you're fine. 
Several times when my available credit was high enough I've done this on other cards to lower my own interest rate on a balance I was carrying. (I mention my available credit being 'high enough' since in my case I needed to have a large enough credit line to cover my existing balance plus the BT, since it takes a few days for everything to post and during that time both my existing balance and the new BT were reflected on the account.)
The 'caveat' with Discover is that while you're carrying a 0% or low APR BT, you can't use your card for new purchases without incurring interest charges on the purchases. For most cards (like my BBR) that's not a problem, but Discover can be lucrative for many people for their 5% cash back categories, and there can be a true opportunity-cost to putting the card in the SD. You basically have to determine if the savings from the BT is worth missing out on the 5% categories until the BT is PIF.
Right after I got my own Discover, I did a BT so my 0% BT offer ran concurrently with my 0% purchase new account offer, which worked out well since I was able to continue using the card for new purchases without being charged interest. 
Edited to add: While they are clear that new purchases will incure interest, I'm not sure how your payment would be treated as far as your existing purchase balance goes. It's worth reaching out to a Discover CSR to get the answer on that, otherwise the entire exercise could be 'moot'.
To help clarify, this is from Discover's site:
