How to choose sampling rate for non-band limited signal given as an expression?
It depends on the spectral distribution of the signal under concern. If, for example, as you said the signal has most of its energy around zero frequency and gets progressively smaller as frequency increases then it may be possible to sample it with an acceptable error, provided that the progress is rapid enough. So it depends on the progress: Provided that it decays fast enough, you may find an acceptable sample rate which would provide a bounded error. After all many practical signals will have nonzero (but very small) energy above their half the sampling rate even after aa filtering, but nevertheless they are sampled with sufficient accuracy.

And your bit resolution vs stop band attenuation concern. That is the noise floor of the N-bit representation. Any error less than noise floor, and being uncorrelated with the signal, may be considered ok.

Finaly please note that, when the aliased frequencies fold up (say at 0 hz) for every next shifted spectra to be added, the folded tail will be progressiveley smaller, as it comes from progressiveley farther away from its center. Hence the sum of such infinitely many decaying terms may converge. Provided the folded terms are getting smaller and smaller.