Hit By Pitch
 

ASA 8.1.F
When a pitched ball not swung at nor called a strike touches any part of the batter's person or clothing.
Effect: The ball is dead. The batter is entitled to one base without the liability to be put out.

NFHS 8.1.2.b
A batter is awarded first base when:
b. a pitched ball not swung at nor called a strike touches any part of the batter's person or clothing.

Must the batter/offense accept the award?

Situation:
R1 at third. Best hitter at bat. Not as 'great' a hitter on deck. HBP. Must the offense accept the awarded base?

I can't find any rules in either code that force the acceptance; conversely, I can't find any that allows the offense to refuse the award. Granted, an intentional walk could certainly avoid the scenario, but does the option exist fo the offense?

The award is just that. An award. He may choose not to accept it by leaving the field of play and being called out for abandonment... But to answer your actual question (may the batter continue batting?) - no... he may not.

An "award" is one way a Batter becomes a Batter-Runner; and there is no undoing that.

Can you point me to something in the rule book that states that? I'm fairly certain that is the intent, but if you notice the rule cites above, it doesn't say that the batter is awarded first base, it says they are entitled to one base. Also, there seems to be a glaring lack of a definition of what the rule book considers an award; meaning must it be accepted? I'm not trying to be difficult, just trying to determine if there is a gap in the rules that needs to be closed.

ASA 8.1, A batter becomes a batter-runner when ....

As CecilOne indicates, that means they are no longer a batter. Stay in the batters box and bat, and they are now batting out of order. Cannot call out for abandonment under 8.2.D without the BR entering the team area.

Bottom line, you cannot allow it to happen. If BR refuses to go to 1B, eject for making a farce of the game.

That works for me.

I may well be wrong, but I thought that once upon a time in MLB, the batter could choose to stay at bat. Maybe this was in the 19th century. It also might be pure urban legend, though, in the category of "any strikes on the batter are removed when the pitcher is replaced during his at bat."

There is anothe possibility, an urban legend in the 19th Century. :)