I know this is an old thread, but I have been getting a light gun cab setup with MAME and trying to get this game right, and I think I have found what is a "better" solution via the MAME Input Macros where you can get actual one shot offscreen reloads.
First in MAME before loading any games, you have to enable the Input Macro plugin.
Once you do that, reload MAME and when you load up Invasion you can see the plugin folder in the MAME menu, and click that, and then go to Input Macro and create a new one.
I have a screen shot of my settings below:

And then here is a screen shot of my P1 controls, which are SLIGHTLY different than the other ones posted above. The main (and only difference I think) is that P1 Button 1 is ONLY set to Joy 1 Button 0.

On my gun, Joy 1 Button 0 is on screen trigger pull, and Joy 1 Button 1 is off screen trigger pull.
So basically with the input macro, when you shoot offscreen, the game input settings move the cursor all the way to the right side of the screen.
Then the input macro is also activated, which waits 3 frames, to do a P1 Button input, which is what then shoots the edge of the screen and gives you the reload.
Here is a video of it in action -
https://www.youtube.com/shorts/qtiiUPablrwNow you will notice that like once or twice in that video, it appears I shoot off screen and it doesn't reload. I am not sure but I have noticed it happening multiple times in that one part of the game where a lot of stuff is going on. So I don't know if it could possibly be because of a frame drop or something.
I have also messed around with the "frame delay" option in the macro settings and when set to 2, it doesn't activate that often but sometimes will, and 3 seems to active like all the time. I will say you can also FEEL that slight 3 frame delay of it happening since it's not instant like reloading in other games, however I think it is more natural and easier than having to do the double tap to reload, atleast for me it did when I was in a hectic part of the game.
I also had this working with the auto-fire plugin, but with that you had to hold the trigger down ever so slightly when pulling the trigger off screen, and it also felt just not as natural as the single click we're all used to.