Is awarded base mandatory?
 

HS varsity game. Score was V4-H2 beginning the top of 7th inning. Visitors get a rally going and eventually score 12 runs in their half inning -- several good hits, some lucky hits, some errors by defense -- momentum is just snowballing for visitors! Eventually both teams are looking for the 3rd out to get that half inning over. Near the end of the rally, with the bases loaded, BR hits a blooping fly ball down the right field line. F9 dives, doesn't catch the ball, her glove hits the ball and it rolls OOB and into dead ball territory. A couple runs score, BR gets to 1B, rounds the base then comes back to 1B making no attempt and showing no interest in advancing to 2nd, as though she doesn't want to add any more embarrassment to the home team. Her coach says nothing. Do you stop the game and force her to go to the base she is entitled to or let her stay on 1B? Is it mandatory that a player accept the "awarded" base?

Absolutely!

Home team comes up in their half of the inning and scores multiple runs now making a "game" of it. Suddenly you have the same identical play, but this time the player goes to second. What are you going to tell the visiting coach when he comes out to question why his girl only got first?

This may be uncomfortable for both teams, but let them know the rules, award the proper bases. Its the only way they may know what the real ruling should be!

More than that, a rule is a rule. The player is not embarassing the opponents by following your directive and there are other ways to keep the score down.