How should I handle greetings when unable to bow, without seeming disrespectful?
You don't, you either bow or you disrespect their practice. The reasons don't apply.
In much the same way as declining a handshake for any reason beyond you missing a hand is disrespectful, or refusing a Maori hongi at a marae is disrespectful (don't do this last one, it's a direct challenge to someones mana).
However many nationalities are used to disrespect from foreigners and give them leeway, so it's not something you need to worry unduly about in most cases especially in a workplace. Unsure on Japanese, they were very respect conscious when I had dealings with them, but with many nationalities, it's fine to be a foreigner with colleagues in these matter, but it can work against you with superiors although they may never stop smiling.
For example, in my own country any perceived disrespect to superiors could lose you a contract even if you were by far the best candidate or price. Yet I didn't grow up here and my interpretation of my religious prohibitions have some fundamental differences to those here even within my own denomination, in which case I disregard their norms for what I believe. I pay the price of being perceived as disrespectful by many which I understand because I am consciously disrespecting them. This has in pragmatic terms lost me many opportunities and is one of the reasons I started my own business rather than continued working for others here.
It's outside the scope of the question, but I would advise asking your co-religionists or leaders that have already encountered this issue how to handle it in a religious context.