Without googling the exact language you are only required to produce a stamp when requested by the ATF, not produce one immediately, I never carry a stamps. If you ever have an ATF agent is asking to see a stamp I'm willing to bet you are about to be properly f***ed for something much bigger. I bought my first NFA item in 1994 and I've never interacted with anybody the ATF asked to see a stamp, it practicality it just doesn't happen, unless you messed up bad.
That said, if you want to carry a stamp you only need to carry a copy of the stamp page, that is your permission slip. The other option is to set up files on your phone or on a shared drive with all of your stamps. On a shared drive you will have access on all of your devices. I have a two main folders, Form 1s and Form 4s. Form 1s are just a bunch of SBRs I've made. In the Form 4 folder I have sub folders for SBRs, Silencer, Machine Guns and Destructive Devices. Then I drop the stamp in the proper folder.
The only exception I have is when I fly with an NFA item I print a copy of the stamp to shove in the case. This is my built in defense against the potential of an overzealous TSA dude. I have never had an issue with TSA to be fair.