This could be complete twaddle, DJ, but when I send / receive faxes on my PC (I know yours is a separate fax machine), it uses the standard 56K modem. If this is connected to the phone line via my broadband filter, I lose my broadband connection each time the 56K modem is used. I only found this out when I moved recently, and had inadvertently connected it in this way. If I use a phone socket doubler, so that the 56K modem is connected to the BT line separately to the broadband filter, everything is OK.
Maybe you just need a similar set up, so as to connect your fax machine directly to the BT line, without it going through the router?