The crucial tidbit of information I was missing is the correct parameter name token_index as described in:
When this token_index is out of range, you’ll receive an HTTP 400 response code with the body {"reason":"Invalid request : Maximum group number limit exceeded","error":"invalid_request"}.
Update (27 September 2022): Developer Support has explained that the number of simultaneous token index values is limited to 3 concurrent tokens due to the “higher security risk of having more index token [sic]. Our back end can’t increase it to more than 3 index token [sic] without a genuine business impacting use case, considering the security.” I’m still getting the group limit exceeded error message described earlier when trying to use any of the additional token_index values above 0, so my personal recommendation is to save yourself the trouble of working with Developer Support and just create duplicate identical applications for however many concurrent tokens you need.