Is this on a US-based airline? If so, FAA policy is that you may show the relevant portion of the manual instead of the sticker if the sticker isn't legible, but it sounds like you knew that already.
FAA Advisory Circular 120-87B, Section 11, Worn or Unreadable Labels,"
"An owner’s manual is also acceptable as proof of safety standards, as these booklets contain pictures or illustrations of the CRS and information that the CRS meets FMVSS 213."
There's no additional detail there, so it's hard to say what the minimum is in terms of documentation. But if it were me, I'd make sure to have both the cover page of the manual -- so that the carseat can be identified -- plus the page(s) that show aircraft installation and approval for use on aircraft. In other words, I don't think there's any intent to require showing all 22 pages of the manual, since most of them are irrelevant.
Whether you want to do this in hardcopy or electronically is your call. I think hardcopy does tend to be a little easier in these situations -- you can get it down to a single sheet if you print front-and-back, and that allows you to just hand it over to the flight attendant if there's any question, instead of everyone hunching over the phone and scrolling through electronically.
I think I'm asked for the sticker on maybe 10% of trips, but people's experiences seem to vary quite a bit, so it's good to be prepared.
