Although two wireless cards in one PC will have some interference issues, it is still possible.
I have tested a laptop with two wireless connections, to two separate wireless routers and both work - (on different channels) and the second one connected (a USB wireless device) connected and obtained its IP from the DHCP on the second router while the first wireless connection was active. Both report 54Mbps connections although I can't test actual throughput. Pinging both wireless NICs from a different PC suggested that there was some delay when both wireless NICs were active, but only on one NIC, so the issue may be in Windows OS rather than a wireless contention.
As a result I think that the basic concept of the bedroom_1 PC with two wireless connections is workable. There may be some interference and reduced throughput, but it is better than no connection. I can't test throughput rates on my setup.
It would be better for the bedroom 1 PC to have an ethernet and a wireless connection, but I don't think that this is possible for Sidious 999 from what has been posted. I would be happy to be proved wrong on this. Hopefully Sidious 999 will tell us whether an ethernet hardwired connection to the Internet is available in bedroom 1.
Sidious 999 said that no new hardware was possible, so repeaters/AP's are out.
Apart from the 'no new hardware' issue, a repeater could be placed in bedroom 1, and both the PC and the MAC could connect to it. A cheaper home wireless router could be used instead, but the throughput falls due to the single channel connection. I don't know enough about repeaters to be be able to comment on this setup.
In the configuration I suggested as possible, Windows provides the bridging between the two NICs, and AFAIK there is no reason why two wireless NICs won't link (but I have never tried this setup, so I may be wrong!)