I know it is a bit more work and mess, but I am very skeptical about how effective this process could be. I can't help but think it is comparable to this method for adding engine oil:
You can tell this young lady in her DIY jeans really knows what she is doing, because she is smart enough to make sure she is getting the oil evenly spread all of the engine bay. #doingitright
Think about it, if brake fluid (at around a 1000 PSI) is not getting past the seal, how is grease at 1 bar going to get in there and lubricate the seal? It is about as likely as the oil these ladies are pouring all over the engine getting inside the engine.
Just get a low wide bucket or plastic bin to place under the caliper, a couple cans of brake cleaner, and some shop towels, and it really isn't that big of a deal. Honestly jacking the car and taking the wheels off and on is the hardest part IMO, so if you are already doing that I say take a few more minutes and do it right, there are far worse things than a little spilled brake fluid.
The easiest way to do it is to remove the pads and use the system to push the pistons most of the way out with the brake pedal. However without a pad the pistons will push past the seal so you can end up with one piston out with a brake fluid mess and the other three still more or less recessed. So you want to put something thinner in place of the pads first, a completely worn set of pad backings would be perfect if you have them, but any bit of wood or metal about 5 mm thick will do.
Once you have the pistons most of the way out you can use a stick to depress the brake pedal and disconnect the line. Then if you have compressed air you can push the fluid in the caliper out through the line/bleeder screw (this is optional and not necessary). You don't have to completely remove the caliper to do this but you were already down to a single loose bolt so I highly recommend removing them, just drop it into your shallow wide bucket and clean it up on a bench/table.
You want to clean the seals with alcohol/paper towels so they are not covered in brake fluid before applying the grease. So just clean everything up first then grease it up really well and pop it all back together. It really doesn't add much time and effort to what you were already doing, and if you have a bucket and some cleaning supplies it is not bad at all.