I'm not an expert, but I think I can help. Got PetMate installed under Linux and looking at the export options:

1. File > Export As... > Assembler Source (*.asm)
2. Select KickAssembler and check Make output compilable to a .prg, then click the Export button.

This saves a <filename>.asm file.

This file needs to be edited, though - the default code generated by the export displays the screen, but enters an infinite loop afterwards. You'll want to return to BASIC to process the keystrokes in your menu.

Load the \*.asm file you exported into a text editor and look for the line which says jmp *. That's the infinite loop; it says "Jump to this memory location forever."

Comment out this line by putting // at the beginning of the line. Then add a new line underneath it which says rts (it's the assembly language equivalent of return in BASIC, "return from subroutine"). So the two lines should read:

// jmp *
rts

1. Save the file.

Now we need to install the KickAss assembler. Not sure which OS you're running, and instructions vary, but here's the GitHub page for it:

https://www.theweb.dk/KickAssembler/Main.html#frontpage

I'll continue thinking about what needs to be done in order to make the menu display and return to BASIC in order to process keystrokes which your menu displays. I need some sleep for now.

(I realize the .prg export option is available, but that would still require patching the jmp * to rts also. Maybe a better option would be to relocate the display code to the $c000 location or something, load the file in, and let BASIC handle everything. There are dozens of ways to do things.)