This. Once a week I leave the razor to soak briefly in hot soapy water while I give my brush a bit of a clean, then I come back to it and give it a once over with a toothbrush. It's shiny and clean and ready for the week ahead.
I mostly just rinse mine when I'm done so there hasn't ever really been soap scum stuck to it.
I've used IPA to clean/disinfect, it's fine.
If in doubt, test it on a small spot first.
What type of metal is your piece? The best cleaning routine will depend on this.
I've got a Wolfman WR1 in stainless steel. I've never had to do more than disassemble and buff with a microfiber towel, but I have no ides why a normal steel cleaner like Barkeeper's Friend wouldn't do the trick.
But you shouldn't need to do this every shave - if your lather is good, you should be able to rinse it with water, and it should be gone. Otherwise, your lather is too dry.
I use hot water to rinse the soap off the razor as I'm shaving. There might be a little bit of residue that builds up, I try to hit it with some hot water. If it's my straight razor I sometimes carefully wipe the blade with a towel before stropping (this helps dry it and prevent rust, as well as remove any remaining soap scum).
On DE safety razors I take a q-tip and sterilize the blade with 70% isopropyl alcohol which happens to also clean the soap scum off. I sterilize before shaving to help prevent infections to the follicles that can be caused by bacteria that start to live on your razor. Another recommendation I've heard is just to store your razor in 70% alcohol, but I prefer my way.
I rinse mine under the faucet, take the blade out and wipe it by pinching it between my fingers and wiping perpendicularly and AWAY from the cutting edge while holding it under running water. If you're not comfortable with touching the blade I'm sure an old toothbrush can do the same. Then I reassemble it, swirl it in a container of isopropyl alcohol, shake the excess off and set it to dry.
I have found that the last step with iso alcohol makes the blades last much longer. Probably because it helps prevent corrosion of the cutting edge.
Like others said dish soap and toothbrush after every shave but for your current dilemma you should add what underlying metal and finish it has to get the answer you're looking for