The practical upper bound on the serial port speed is not dependent on the overall system speed so much as it is on the quality of the UARTs used in the serial port equipment, although both ARE relevant. Common UARTs in the 1980s were typically usable up to a maximum of 19200 baud in systems from that era when connected to external devices. So you can probably use any serial port on any machine for your purpose as long as you cap the serial port baud rate at 9600 or 4800 baud. If that proves to be too fast, lower it still further (2400, 1200 or even 300 baud).
If the issue is having to run DOS software to program the radio, then you might consider looking for a machine with a 32-bit version of Windows with RS-232 serial port hardware as that should be capable of running your 16-bit DOS program. The properties dialog box for the serial port in Windows Device Manager should allow you to lower the serial port baud rate to whatever baud rate you need for your external circuit as well as set other parameters such as parity, start/stop bits, etc. Obviously, you would be better off it you can test your software on a 32-bit Windows system before you spend any money in order to make sure that it runs properly. For example, your program might require graphical modes that more recent graphic adapters support poorly or don't support at all.