My house is about 456m away from my misses house - we live in the same village...
So yeah, it's really not a big deal, your boss is technically right perhaps but meh...
As Burnsy has pointed out that the law isn't all that clear - the rules themselves seem reasonably clear but presumably, the definition of a household muddies things a bit here and might need to be tested in court... what are they using to determine what your household is etc..
I mean do you have your post, payslips sent to one address and her to another? Do you pay council tax with a single person discount at your address and she pays council tax with a single person discount at her address? Are you on the electoral roll at your address and she's on the electoral roll at her address?
I suspect you probably tick enough boxes that someone could reasonably argue you are two different households and you've formed a bubble together (since August), in your bubble you're living at each other's houses as is allowed, part of the reason for bubbles is to allow for what you're doing with her. To add a third person would technically go against the rules.
I guess if you were to, together, pick one of the two houses, both move into that house (and therefore have your post sent there, pay the appropriate level of council tax for a two person household at that address etc..) then you could claim you're a single household and you'd be able to form a bubble with another household containing a single adult.
So - what is your council tax situation? Are you paying it as two single-person households or have you picked an address where you pay it as a household containing two adults? And another address where you're claiming a second home discount? If two single person households I suspect your boss is right. That would seem to be at least one way of seeing what you two have already self-defined your situation as legally.