SymphonicAlex: P.S. I advise you to buy a 360 gamepad because it works so much better with all games (especially the old ones, since the new ones don't have problems with other gamepads).
I know it's been a few years for some other reasons I'm seeing this, but I'd like to address this one specifically to say that this advice is useless.
An F310 controller is a controller which combines the two worlds of the ergonomics of a Playstation style controller with XBox 360 software. The very fact that the controller has X-input makes it recognized as a 360 controller. It is virtually the same device. Simply buying another one is a recursive solution which repeats the same problem when it comes to this franchise.
The D-input is usually used for older games. It was actually the only option used for the older iteration of the F310, which was virtually identical while only using D-Input, except it lacked the guide button and didn't have trigger buttons modified for accelerators (as it was mainly a button, not spring loaded sensory). It is an option to use for this series. However, with that being said, specifically for BR1 for example, there are caveats.
The caveats have to do with the game itself and the reason I ended up here at the moment. This is not my first foray into games with weird controller schemes. One problem is that the trigger buttons themselves are recognized as axes in both X-input and D-input modes on an F310, and I'd imagine just as well as a problem for 360 controllers, so the point is moot. That is complicated by the fact that because they're used as axes, there is no "Button 12" to use to even shoot. Additionally, the right stick only has 1 axis and that is horizontal. It's used by default for moving the camera up and down, and yet there is no input recognized when you move the stick up and down.
The guide button as well as the two upper middle buttons (for conventional back/escape use) also generate no input in the game even if you tried to map them.
I have tried other "fixes" using patches to replace the ini files and other .DLLs amended to my install. There was no difference to it. I'm even trying to run it through Steam (which I usually do for non-Steam games) and the only thing that I've managed to do thus far is use one of the middle buttons to press Esc like modern games use. Other than that, BR appears to have been made for a different type of controller in mind at the time it was made. This is not to say there isn't "full controller support", but it would be a very loose term in this case because it is not compatible to modern standards (which again, isn't a new thing).
The only other alternatives I would try would be to use a third party mapper style to use keyboard inputs on your controller with Steam or a third party mapper. It is irrelevant to use any specific controller at this time, unless you have one that was made/compatible for the game at the time (which would be great!). Any modern controller is still going to run into issues with legacy games these days.
EDIT: Also, even using the 360 controllers, you do in fact have the same issue as reported
here in the comments. So again, buying one is not gonna solve the problem like one and done. Otherwise, again, you have to adapt or go back to the KBM controls.