No. Asking the agency a question does not pause a response deadline that is already running. Plan the six weeks around the dates on the notices you hold, arrange remote signatures in advance, and remember that official descriptions of processing in weeks are descriptions, not a guarantee of any date.
Working a calendar around an absent applicant
List every dated obligation, then work backwards to the last day the applicant can realistically sign. Track the authorized portal, mail and representative communications required for the case and never hand over credentials so a deadline can be met. A pending case and a paid fee are not work authorization; permission to work generally follows admission as a lawful permanent resident, unless the person already holds independent authorization.
A currency transaction does not itself change the official processing stage or create priority. The applicant pays the nonrefundable processing fee, undergoes vetting, and only after successful vetting makes the gift. Nothing about urgency, a closing window or a limited allocation justifies departing from the official channel, and a promoter who resists that check has answered the only question worth asking.