Personally, I would do this:
Buy bleed kit. You can get them on eBay for a lot less than the official one. While waiting for this,
Take out pads & clean out any crud from the edges of the pad so they are ‘clean’.
Push back the pistons & clean the caliper/piston with water & a small brush/rags.
Squeeze the lever a few times while observing the piston to see what they do. Don’t squeeze them too far out or the piston will POP out completely. The pistons should move in evenly from both sides.
Push the pistons back in & stick the pads back.
Watch the Avid Elixir bleed video on YouTube a few times. I make notes at each stage so it’s easy to refer to.
When the bleed kit arrives, methodically bleed the brake. Make sure you attach the syringes to the rear lever & the rear caliper. Twice now I have wondered why I can’t get fluid through the system before realising I had put the syringes on the front lever/rear caliper.
If you bleed the brake & you still get the problem, you probably have a seal problem.
It’s easy to bleed the brake following the correct procedure & taking your time.
I wouldn’t put any types of cleaning solvent near the caliper as it may cause damage to the seals.