220 to 110 in subpanel with no neutral
In that pico-panel, you have a major defect. You have the breakers incorrectly swapped.  You have a 40A breaker on the thin wire and a 20A breaker on the thick wire.  The thin wire is totally unprotected.  Swap those breakers!
You are correct.  Without a neutral, that pico-panel cannot support any 120V loads end of subject.
You cannot use UF-B as extension cord and I cannot imagine what you were thinking there.   Being solid wire, it is not flexible like a cord must be.  It is not designed to be flexed over and over.  You need cordage such as SJOOW.
If one 10/2 UF-B is run in conduit the conduit must be 3/4&quot; or larger.  Nothing else can be in the conduit.
Honestly, I think your best bet is to run the proper and final feeder that you ultimately want from your main panel to the shed.  I advise against having the power company drop you another service, because the power company has a new trick: they are charging people a &quot;minimum provisioning charge&quot; which, if you cancel service, becomes a &quot;service availability fee&quot;.  That is, they charge you for the fact that you could have a service drop if you wanted it. I now have 2 sites with 2 PoCo service drops each, and they charge us $60 a month per drop no kidding.  They won't let us out of either one.  So needless to say, if I have to haul power 1000' I'll feed off my existing service rather than order a new one.
Service size has no bearing on it; one is 240V/100A and another is 480V/1000A. Same $60.