If you want a satin finish, there's no reason you can't achieve it with an oil based satin poly. You will likely need to sand the back & sides down to bare timber again, as trying to apply over the existing tacky stuff is just asking for trouble IMO.
I would really encourage you try my 50/50 mix of satin poly and mineral turps (aka white spirit in your country) and wipe it on.
You can also go 60/40 poly/white spirit for the early coats to get a bit quicker build, then go 50/50 on the last few coats for a smoother more flowing consistency.
If you choose Tru Oil for reasons of familiarity and consistent results, you can de-gloss it with wet sanding to have a more satin-like appearance. However it will not give a true satin finish because real satin clear coats contain a small proportion of solids that diffuse light (minimise reflection).