This amendment is not germane to the intents of the bill and it is inappropriate that it was ever considered. The VP should have rejected this amendment. If this is your intent, reject this legislation and propose a new one - this is not how proper governing should be done.
The House, as I am aware, does not have any rules regulating content of bills beyond frivolity; certainly not anything to do with the "intent" of content. As has already been mentioned, the precedent is in favor of allowing this sort of thing. If the House wants to change the precedent they may go ahead, but as VP I feel I would be overstepping to impose a content restriction unilaterally.
Also, I must warn. Going ahead and putting such a restriction in place would lead to a massive mess of personal interpretation and blame-pushing regarding what exactly counts as "initial intent". Some members might argue, for instance, that even
weakening a bill in order to let it pass as "against the initial intent". Or changing just one subsection in a large bill into the opposite etc.. I don't see of a good way to implement sich a rule without making everyone's life miserable