Script presses Ctrl (default fire button) 8/sec whenever Ctrl is held. Works only when "DOSBox" window is active. I arbitrarily chose fire rate since it felt human but you will regardless be limited by bullets on screen the game permits (1, 2 with upgrade, 3 in one level). Functionally there is no benefit except saving you a surgeon visit after your little pinky inevitably falls off.

Essentially it's a .txt file in .ahk format (AutoHotkey script), compiled into standalone executable so it can work without having AHK installed.

1. Download standalone executable[1drv.ms]
   
2. Run executable when playing HP
   
3. Re-define Fire & Jump controls to Ctrl & Space
   (Change game options > Define key controls)

To terminate script execution simply shut it down.
Last step aligns controls with default ones fresh off the original disk, for sake of compatibility.

Increased size is because it also contains AutoHotkey interpreter.

If it still doesn't work, rename "dosbox.exe" to "DOSBox.exe". Detection is case sensitive.
Believe me I tired to get around that but manual is hostile to noobs (my 1st AHK script ever).

Tested only on 64-bit Windows OS. No clue if this works on Linux or Mac. Sorry.
You can try re-compiling it in OS-specific AutoHotkey with content directly above.

VirusTotal scan for concerned Paranoid Androids: HERE[www.virustotal.com].
There are 5 false-positives as you can see. Probably a medical cartel conspiracy.