This exact situation is covered in the USBC rule book.
If the Sub. was not sanctioned at all or just needs the local assoc. card. According to USBC Rule 101 he has until the end of the series to pay the required sanction fees.
If they are not payed by then, The team forfeits the games the sub bowled in.
Since the sub didn't fill out a card and pay the required fees till after the end of the series.
The team the sub was on forfeits the games.
The other team in this case had been bowling in what is called a "unknown forfeit" and is automatically credited with all of the points.
The situation is covered under Rule 110 & 119 see the CAQ below
USBC Rules
Rule 101 – Membership Fee Payment Requirements
To be eligible to bowl in a USBC league, a bowler must:
a. Complete an individual membership application in each league the bowler participates in.
b. Pay or show proof of payment of USBC dues and the required association membership fees in effect for the current season before:
1. Adult membership: Completion of the bowler’s first series.
2. Youth membership: The bowler’s third session of competition.
Any team using a player who has not satisfied USBC membership eligibility requirements subjects the games in which the bowler participated to forfeiture.
Commonly Asked Questions – Rule 110c.
110c/2 What is an unknown forfeit?
The team bowls unaware that a forfeit will be declared. For example, it was found out immediately after bowling the opposing team used an ineligible player. The forfeiture would have to be acted upon within the specified timelines of Rule 119.
The forfeiting team does not receive any points and the non-forfeiting team is automatically credited with all of the points.
110c/6 If a team has to forfeit because an ineligible player is used, what happens to the scores bowled?
An ineligible player’s score becomes zero, and the game(s) is (are) not included in their average. The team’s score for the forfeited game(s) is (are) also zero. However, the scores bowled by the eligible players are counted toward their averages and any individual prizes.
119/2 When a team uses an ineligible player, can a league officer declare games forfeited?
Yes, league officers have authority to take action regarding games bowled by an ineligible
player if detected within the time frames stated in Rule 119. If action is not taken within the
time period, the games stand. League officials have the power to enforce league rules and to
declare game(s) forfeited. If an officer fails to act, or a league member decides to challenge the
officer’s decision, a written protest would have to be filed in accordance with Rule 119.