Well this is just a stab in the dark here. I don’t own a valves instrument but I do use lubricant oil. Would a needle applicator help? That way you can be precise but don’t have to worry about spills

Maybe a super absorbant hand towel you can set on your lap or on the floor before oiling the valves? Shop towels also work well on oil, and you could stick a few sheets in your case. As a horn player, I always keep a hand towel in my case for inevitable oil, slide grease, or “spit” drops.

I mean, my trumpet has a cloth with it, not really sure if it was for this or not but I usually put that on my lap when oiling to catch any spillage and whipe my hands or the trumpet off with it afterwards. Also, its probably similar with baritones, but I don't take the valves completely out either. I unscrew them and pull them out slightly then oil the tops, the oil will drip down anyway. Whether thats a good habit or not I don't know, but I had a trumpet teacher once show me that, and it seems to work for me as it works for him, hope that helps some.
Edit: Why I mentioned not pulling the valves all the way out is that creates less cleanup as your not dripping oil everywhere, oil should drip down into the instrument and help lubricate further while also not getting all over.

I am thinking if you chill the oil it might become thicker ( like grease ) and it might be easier to tell where it is. Maybe some kind of bottle or tub you can just dip the valves in.

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