OP, you are correct. There is no standard way for any video card in a PCIe slot to communicate its temperature info to the mobo. Hence, there is no simple way to do what you want. There are SOME mobo makers who have included something like this on SOME of their mobos IF you use one of their specially-designed video cards with one of their mobos. Basically they have added a non-standard data transfer method to both their mobo BIOS and their video card so it can be done. In a somewhat related vein, many video card makers have set up a way to send their on-board temperature data out to the mobo in a way that allows the card maker's custom software tools (included in their video card driver and management tools) to display it for you. But that's still not an industry-wide standard, so no mobo can access that data - only custom software designed for that video card can do it.
A third-party fan controller is VERY unlikely to include such custom software to read temp data from a video card and automatically alter a fan speed in response.
HOWEVER, there MAY be a tool you have already. Often (but not always) the drivers provided with your video card also include a card management utility you can use to observe the card's operations and adjust some of its settings. IF that card includes a cooling fan system, many such cards actually apply their own automatic fan speed control system on the card (rather than simply having the card fan run full speed always). IF your card has such a feature, you can use the card's management / configuration utility to observe the cooling operation and GPU temperature, and MAYBE to change its operation to run higher cooling that it normally does. In other words, adjustment of the video card's automatic fan control, IF POSSIBLE, would be done NOT in BIOS Setup on the mobo, but by using the video card's management utility.
By the way, I generally consider that case ventilation fans' speeds should be controlled by the SYS_FAN or CHA_FAN header by using the temperature sensor on the MOBO, not the one inside the CPU chip. I don't know if changing that will actually help your situation, but it is better for the majority of mobo components.
The only other non-standard way I know to address this issue involves custom re-wiring that may void your video card warranty and may NOT be possible, depending on what your hardware is. On some mobos that DO have thier own fans AND do their own automatic control of them based on the GPU chip temperature, the connection to the on-card fan is accessible in some way. For most of these, the fan used appears to be very much like a common 12 VDC 3-pin fan that uses varying voltage supplied to the fan motor for speed control. So conceivably you could tap into that output to the card's fan and splice in a connection to an external case fan. BUT that depends on whether the card's fan power supply can actually handle the added load of an extra fan.
A simpler option, although not automatic, may be to use your mobo's configuration utility tool supplied as a Windows app so you do not have to be in BIOS Setup so see and manage the fans. There you may see your mobo's fan headers and be able to alter their settings. If you connect that one side fan to its own mobo header, you can have manual control of its speed, rather than allowing the mobo to use its automatic control function. THEN if you use your video card's configuration utility to monitor its GPU chip temperature, YOU can be the controller that sets the side fan speed manually as you see fit.
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