Well, *most* DOS programs with mouse support access the mouse through the standard INT33 mouse driver interface of DOS, which is the same no matter what kind of mouse (USB, PS/2, serial, bus mouse) there is. This driver interface is what's created by loading the MOUSE.SYS or MOUSE.COM program in the CONFIG.SYS or AUTOEXEC.BAT of a real DOS. This interface is fully emulated by DOSBox.
The one important exception is DOS programs that support more than one mouse; because the INT33 interface handles only one, such programs usually deal with the second mouse by directly accessing the serial port (serial because no PC I know of had or has more than one PS/2 mouse port, and of course USB mice didn't exist yet during the DOS era).
Does your DOS program support two or more mice connected to one and the same PC? If it does, that's the likely culprit. If it doesn't, the likelihood that it wants to talk to a mouse directly (rather than through INT33) isn't all that high. In the latter case, I'd suggest thinking about whether your program does anything else through the serial port; for example, does it support connecting to a serial printer, or some dongle, or some kind of special hardware, or via modem to another copy of it running on another PC, or to some kind of server?