We for a while did daycare pickup by bike, but our kids were older. 4 months does seem very young to ride on a bike--you might need another car for this year at least.
Once baby is old enough for the bike, a bike TRAILER, as opposed to a seat, is actually very cozy. If there were deep snow or super-cold weather, would you be able to go into work late a day or two? If there is any flexibility on that score, you may be able to make one car work. I took the kids in the bike trailer as cold as 14 degrees and with a modest amount of snow. (With studded bike tire--those things are great!)
We currently have a second car to make life easier--I have to pick up Little Brother from preschool, take him home for lunch, then get him to daycare and myself to (bus-able, but non-bike-able) work within a few hour period. Not practical without a car and Mr. FP needs to have a car in order to make 4:00 afternoon pickup for Big Brother. Next year, Little Brother will be in full-day pre-K and I might sell my car and just take the bus.
Congrats on new baby, BTW!