So we know that there are COVID cases that only have gastrointestinal symptoms [nausea, vomiting, diarrhea]. Presumably, the virus only managed to infect gastrointestinal cells, either due to primary exposure site or innate immunology peculiarities. When these patients recovered we assumed that they are now immune to COVID.
We are primarily concerned about the effect of COVID on the respiratory system. Is it possible for us to give variolation in the form of a small, precise amount of virus in a pill that is only released in the small or large intestine? 
Sure, you would have vomiting, diarrhea, fever but it would be much safer than respiratory tract variolation. Dehydration is a risk but it can be fixed much easier than whatever COVID is doing to the respiratory system.
if we are concerned about the virus contamination even then [e.g. when you vomit or fecal contamination], would intra-muscular injection of live virus work? if not why not?

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Eyes and ears  come to mind. The eye might bring it dangerously close to  the brain.

Rubbing it into a non bleeding scrape on the skin is also possible.

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If you variolate without the respiratory tract you likely won't get mucusol immunity which results in less immunity to getting infected then someone who's normally infected.

We also suspect many asymptomatic cases. So, your presumption that the virus only managed to infect gastrointestinal cells is insufficiently founded. It is perhaps as likely that the virus infected both gastrointestinal cells and respiratory system but was not successful enough to cause respiratory symptoms.