Solved
The instructions for backdoor conversions trigger the contribution limits, so they don't work for this situation.
Code G indicates a distribution that is not includible on Form 8606.
If in addition to having code G in box 7 the Form 1099-R has the IRA/SEP/SIMPLE box marked, the form is reporting a rollover from a traditional IRA to a traditional account in a 401(k), 403(b), 457(b) or the federal TSP, not to a Roth IRA. If the IRA/SEP/SIMPLE box is unmarked, the form is reporting a distribution from a 401(k), 403(b), 457(b) or the federal TSP and distributions from these types of accounts are not to be included on Form 8606 because they are not distributions from IRAs.
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