While the OP is right that the WTS has a permanent eye kept on its tax-exempt status, there's many more reasons that that.
1) As long as if the WTS kept reasonably clear of politicking, its status (in the US at least) would be unaffected. There's no connection between this and whether its adherents are apolitical or not in their personal lives - the organization's tax status would be unaffected.
2) The more time and money spent on politicking, the less time there is for kingdumb proclaiming (i.e. vectoring this awful disease to new victims) and the less money there is to be squeezed into the Watchtower's coffers. This is a more important reason.
3) Likewise with charitable giving or work by its duped adherents.
4) There is an additional factor, and its importance should not be underestimated.
Any loony cult (and there's no big one as loony as the WTS) needs some DISTINGUISHING or IDENTIFYING MARKERS making it stand out - preferably ones that are unique to itself. There need be no logical or rational reason for them. Indeed, the more arbitrary and stupid they are, the better - this is because of the EMPEROR WITH NO CLOTHES syndrone:.
"No one else but we are smart enough to see why God prohibited vaccinations, oops transplants, oops blood transfusion, oops whole blood, or to ban birthdays, or any hundred other crazy senseless idiotic rules - perhaps even soon to ensure that pet cats are theocratically-inclined," etc.
Gives the dubdubs a sense of superiority, and many are far too stooopid to realize they too don't understand the reason for the rule themselves..
"... who are always learning and are never able to come to the knowledge of the truth. " - Aramaic Bible in English, 2 Tim 3:7
So - in common with many other cults - politics is almost taboo.
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Focus
("Kill the Faithful & Discreet Slave" Class)