Modem vs Fax Machine (not playing together nicely)

This problem may be unsolveable without buying a new printer/fax, but here
goes:

I have WinFax on my computer (running XP Pro). It shares the same fax line
as a freestanding fax machine. At night, the fax machine picks up incoming
faxes, because I don't want to leave a computer running all night. During
the day, I can shut off autoreceive on the fax machine, and WinFax will grab
the faxes. But, when I send with WinFax, my own machine ends up receiving
the fax. It's a Brother MFC-7150. My previous fax machine, an HP OfficeJet,
behaved nicely in this situation - it let the outgoing faxes go without
interfering.

Other than messing with unplugging wires, or using a switch, is there some
sort of device that will cut the Brother machine out of the loop when it
senses that the line's been picked up for dialing by WinFax?