I had written a long answer explaining how to "incorrectly" apply your custom printer profile to images for printing. But I felt it was also necessary to add a note to the end of my answer... then I decided to delete everything except for the note.

**Note that the end result is really no different from using your printer profile for soft-proofing during editing (and converting to/embedding the correct color space during save as normal), which is the correct approach in a fully color managed scheme. You said that soft-proofing would make the edits "eye-calibrated;" well, they are adjusted by eye/to taste either way...

Edit to add: This is how I would do what you want using Photoshop.

• First, create your custom printer profile.
• Then open any image and record an action of: Edit > Convert to Profile (your custom profile) > Edit > Convert to Profile (sRGB). Use whatever conversion preferences you think are best at both conversion stages (I would avoid absolute colorimetric and saturation modes). Name the action something like "Convert for crap printer".
• Close the image (no need to save the edit/conversion if not needed)
• Now open: File > Automate > Batch: Select your action as the one to play (will default to last used/created/edited). Select the folder with the files you want to convert. Optionally set a new output folder with a distinctive name. And rename the files with something distinctive to identify them as broken. Run the batch conversion...

The only reason for applying the second conversion to sRGB is so the printer doesn't have a problem accepting an unknown/bad/wrong color profile (would typically default back to sRGB, or none, which would also be ok assuming you edited them in sRGB). The reason for the new folder and rename is because these files are now no good for anything other than being printed by that crappy printer. If you re-open them the colors will be wrong/off (as if being soft-proofed), and they cannot be re-edited back to original (at least not easily/completely).