That chain of events looks legit
The description of the ring of force shield, in part, says, "It can be activated and deactivated at will as a free action." The ring puts no limit on the number of times the wearer can take a free activation to activate it, deactivate it, and activate it again.
A GM can make a house rule to change how the ring works if it annoys him, but, by default, this is how the ring works. Further, while the GM can impose an arbitrary limit on the number of free actions a creature can take during its turn (see here), unless the GM does impose such a limit on the ring of force shield, the ring can be used until the wearer gets bored with turning it off and on.
Note: This answer assumes all of the links in the chain of events involve typical arrangements of appropriate game elements. The chain of events falls apart if the bow is stored in a securely-locked chest rather than jauntily and heroically—if unrealistically—slung over one's shoulder, for instance.