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Which has a higher chance of happening before you've been dealt your cards? A bad beat or a royal flush? I'm assuming yourself and your opponents go all-in preflop.

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Is there some universal bad beat standard that I'm unaware of?
The odds of a royal flush occurring in a 7 card hand are relatively easy to calculate:
There are 52 choose 7 = 125,423,025 possible 7 card hands.
Of those, 4324 are royal flush hands, so the chance of a royal flush is going to be 4,324/125,423,025 or roughly 1 in 29,000

If the bad beat is 4 8's or better, then you're probably looking at something on the order of 1 in 9000 for heads-up play (but this is a lot more complicated to calculate).