First of all... You should tell us which ped it is. What year?. Mileage?
(but as I remember it it is a Euro... so I am not an expert on them)
And I am glad you are a wrench for a living.
And I am almost surprised that you need help.
But..... here goes.
I have never seen a small engine that has to have a fully charged battery to run.
On most of them the battery really only has 3 functions...
1st... as a large voltage regulator (and that is whether it is alive or dead) to keep light bulbs from blowing at high revs.
2nd... to keep the lights from going too dim at idle speeds at night.
3rd... to spin the electric start (if it has one)
So I can say you can rule out the battery as a source of trouble (dead or alive).
You said the oil pump might be on the fritz... bad news.
Why do you say this?
Has the oil level stayed the same for months?...bad news.
But they don't drink a whole lot of oil either.
But if the oil system failed a month ago... no amount of oil will now "fix it".
You say you removed the head and scraped off carbon... OK....
But did you push the piston to BDC and look (with GOOD light) for scoring and scuffing down in the cylinder?
You could have ruined the ring seal by starving it for oil without getting an actual seizure.
If this is a ped with the exposed head and cylinder.... pull the cylinder right off (easy as can be) and take a close look at the piston skirt and look for heavy scuffing... and even aluminum from the piston smeared over the rings.
If that is the case you need an oversize piston and rings and a bore job.
And toss the oil pump and premix it.
Also... put the piston in the cylinder and stick a feeler guage in between them (the piston should be in the BDC position while you do this measurement)....What is the clearance?
Also... You said you cleaned the carb... That means you used 100 psi of compressed air thru all orifices many times?
While you have the cylinder off play with the rod and look for evidence that the rod bearing or crank bearings are gone.
Push it, pull it, spin it... you are looking for too much friction, stiff spots, funky noises, too much play.
You of course have the proper spark plug?