To pump grease in through the bleeder you need to block the inlet hole (replace the banjo with a plain bolt or install a large nut in place of the brake line) and loosen the bleeder half a turn or so.
I use mineral spirits to wash grease out/off of most things and for a caliper I would then flush with something called Tool and Parts Cleaner (aerosol spray from Canadian Tire - there must be something like it in the US), followed by Brake Cleaner.
I did a couple of calipers last week. I used the old master cylinder to get the pistons moving and in both cases, once the fluid had pushed the pistons against a single brake pad I disconnected the line, drained the caliper, removed the pad and put a piece of 1/8" flat steel between the pistons and the caliper, then applied air to the inlet to move the pistons into contact with the steel. After that I removed the steel and pulled the pistons out by hand.
Details & pics here
https://cx500forum.com/forum/build-thre ... ost1005093
BTW: I believe that every shop should have the 4 basic solvents
Water: Cleans off dirt and, when used with detergent, some oil & grease.
Methanol: (AKA methyl hydrate) an inexpensive, general purpose alcohol. Used to remove sticker glue, dried fuel deposits (carb varnish), magic marker and a host of other things. Can also be used as gas line anti-freeze, but it costs a lot less in the paint dept.
Mineral spirits: (AKA white spirit, paint thinner, Varsol)(The only difference between the generic stuff and Varsol is the royalty that goes to Exxon for the brand name) Used to remove oil & grease that soap & water won't and to thin & clean brushes when using oil based paint. Good for black heel marks on floors too. With the exception of carb deposits, generally whatever methanol won't remove mineral spirits will & vise versa.
Lacquer thinner: Nasty stuff - if you leave your fingers in it too long it will dry the oil out of the skin and it will become dry & stiff. Removes any oil/grease residue left by mineral spirits so it is an ideal pre-painting final wipe. Removes fibreglass resin from your fingers and can be wiped onto the surface of old fibreglass to briefly soften it when preparing to add filler. Last chance solvent for whatever mineral spirits and methanol won't remove. Caution: will remove most paint so don't spill it on anything.
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