OK, I have definitely been there, done that. This mama has also declared that she is hiring someone to come on the weekends to help, and then we always come up with a solution.
--First, not to argue too much with your terms, but parents don't "babysit" - it's just parenting!! My DH gets very snippy when ever someone says he is babysitting -- we prefer to call it "being in charge" -- teehee
--Second, when he is in charge and the baby needs to nurse, take that 20min or so and snuggle all together on the couch for the feeding -- I think our DH's feel cut out (or isolated) from nursing, so this might help to bring him into the circle of love.
OK, so that said, here are my tips:
-- Try trading off during the day -- for example, if you are a morning person, you can do weekend work 7AM-12PM, then be in charge until 5PM. Spend the "witching hours" (for us that is 5-7) together, you need teamwork at this time, and everyone needs to wind down.
-- Have one place where you do your work, close the door, and have your DH text you or call on the phone if he needs your assistance (AKA breasts). If this isn't working, then leave the house. Or start interviewing a weekend helper.
--Try to spend at least one weekend day as a family, just relaxing. Take naps with the baby. Have sex in the living room while the baby naps! If you constantly have work "to-dos" hanging over you, it will be hard to keep going (especially on no sleep), and your work will suffer.
Things are not perfect in my home, it is an ongoing battle -- especially with an intense, colicky baby and two working parents. It is better now with a toddler, but weekends are still difficult to navigate when I have to do it alone.