Yes, if it has a TI or Cyrix 486, it has been modified, lucky you!

If you are unaware, immediately remove the RTC (clock) battery if it is still present on the board. This will leak and destroy the FDD controller, then the rest of the board over time.
There's a lot to futz around with these cards, and upgrades are another rabbit hole for another reply. (Mine has a Cyrix 50Mhz i486 SLC-2 and math co... just because...)
Anyway, your beep, means the bridgeboard has passed BIOS POST and should display a screen. Do you have a VGA card installed in one of the ISA slots? If so, there should be a DOS screen on a VGA monitor and not your Amiga monitor.
If you do not have VGA (or any other PC graphics card) installed in an ISA slot, then you should use the monochrome or color programs installed with Janus.
I have to run monochrome anyway to get my keyboard and mouse to work, even though I have SVGA installed.
On my amiga, I get garbage in the monochrome display, that is because the bridgeboard is using VGA. For you, there should be a DOS display. Did you enter "PCPrefs" and change the default graphics mode? Try these settings:
Monochrome Display Emulation = Enabled
Default Bridgeboard Video Mode = Monochrome
Reset the "PC" with the "PC Reset" application, then open a monochrome window. You should see your DOS. If that works, then you can try color. You may have to enter the PC BIOS and properly match your display mode.
If all else fails, I'm looking at your virtual HDD partition as the culprit. I use an ISA card since the virtual partition was too slow. The partition is a great way to first set up your card, however, and it appears you did so correctly.
You mention you have an A4000, I haven't tested mine in one of those machines. There are rumors or facts relating to Zorro memory above 8MB and conflicts with the Bridgeboards. I am using an Amiga 2000 with Commodore A2091 and a total of 6MB Fast RAM, 1MB Chip. The A2000 is the best for a Bridgeboard in my opinion, since you can solder the 16-bit ISA slot extensions for a total of three 16-bit useable slots!
Hope this is helpful.
Tim