"He who marches out of step hears another drum." ~ Ken Kesey
Perhaps similar as to how they imprint silicone wristbands
If it's a flat surface look at screenprinting with a 2 part padprinting rubber ink with some retarder added to make it screenable. I do pucks all the time.
As for screen printing, no, nothing from a craft store is going to work, hence the two industrial ink suggestions I posted. Nazdar may have a solvent based suggestion with some "bite". They will even test on your material if needed.Union Process 800 rubber ink is made for adhesion to latex, probably more like balloons. Let's see of our resident balloon printer chimes in.
I also have screen printed with pad printing ink if it is slightly reduced...http://www.inkcups.com/printing-inks/screen-pad-printing-ink/sb/Default.aspxThis will stick to some products that NO other ink will adhere to.