Well, I"ve been playing a 12M for 40 years now as my main horn.
Key to good functioning of the LH little finger mechanism is reduction of friction. There are a lot of linkages with sliding motion. These should at a minimum be regularly lubricated with cork grease, but best practice will be to replace cork with Delrin, PTFE, or UHMW-PE on the sliding joints.
Similarly, because the G#-B-C# linkage involves sliding motion, using those advanced materials will improve the performance of that LH cluster, a LOT.
You can also do a small amount of adjustment of WHERE the G# linkage plate's contact point touches the B and C# keys. This has a small but meaningful effect on the force, since the keys are all pivoted on different axes.
The C# key on the Conn actually is a LOT easier-operating than on the vast majority of vintage horns, due to its two piece construction which makes best use of mechanical advantage. However, once again, you need a little plastic tube on the key at the point of sliding contact, down by the key cup.
Finally, the G# linkage itself is prone to excessive friction due to its geometry, so yet again you need to replace the cork with one of the plastic materials. When you do that, then you can have a lighter G# key spring and the pad will still reliably close. With unlubricated cork, that spring has to be real strong to make sure the G# pad reliably closes.
Last point on slicking up the cluster - as with many other points in the mechanism, you need to decide for each point where sliding motion occurs WHERE you're going to put the cushioning material. Choose wrongly and there'll be a sliding part that tries to plow up yoru cushioning material every time it moves.
And make sure for any linkage joint that has sliding motion and cork, to regularly lubricate it with cork grease.
I assure you that MY Conn 12M has a slicker more positive key feel than any Selmer-type with their enormous key travel, spongy keywork, and the dreadful tilting Bb that falls away from the weakest finger you have just when you need a firm positive key action.