I play an English Edeophone - a top-of-the-range instrument. I don't get a problem with the tips of my fingers, even when I am playing two notes with one finger (mostly when I am accompanying my singing). I remember when I started learning and religiously practiced every day that my fingers ached but it did go away with practice.
Instead, my problem is with the little finger in my right hand. I usually rest the left end of the instrument on my left knee, thus working the bellows and taking the weight with my right hand.
As a result, the harsh metal finger rest digs into the soft tissue of my right little finger, leaving a mark in the flesh and causing my pinkie to get pins & needles! All together now: "aaarh, shame"! The solutions I am considering are to wear a rubber thimble or to stick some tape over the metal....
Keep playing.
"... for he is free to come and play...!"
Alan