It just feels pretty daunting that we have so much to do. All his appointments are pilling up, hearing test, gp, registering, it all seemed so doable a couple of months ago.
I don't understand.
By your description, the work is only answering max 5 phone calls a day, taking a few minutes, and replying to emails.
That's it?
If it is then, a) it's honestly not a job, it's a couple of hours helping out in the business and b) entirely doable.
The issue with small babies / children is being able to be professional (hard when they might cry or need you unexpectedly) and focus on mental tasks (nearly impossible as they need your full attention).
You don't have to do either of these things. Yes, the calls, but these are brief & intermittent.
Would you not consider a more substantial role, designate days to it & look for childcare for those days? That way, when you're working, you're working, and it's worry your while to have childcare. When you are off, you're off.