This is what it says:
Lay on Hands can be cast repeatedly, but cannot apply more than three casts worth of shielding at once.
The salient part being
"[not] more than three casts worth of shielding."
Your proposal:
applying up to a maximum of three times the shielding
This has the same problem, it won't always apply three times the shielding. Saying "maximim" here doesn't really clarify that. It's also worse in my mind. Three times the shielding of what? You're missing that it's three casts worth. I mean, I obviously know what you mean, but I feel like it's even less clear.
The confusion stems from thinking that Lay on Hands can always apply three casts worth of shielding regardless of having other shielding. Obviously not, as you've discovered.
The only way to clarify that is to acknowledge other shielding. Something like...
Lay on Hands can be cast repeatedly, but cannot cause you to exceed three casts worth of shielding.
To me that's perfectly clear and doesn't even need the "at once".