I'm 6'1" but my head never touched the headliner in my Coupe, I could put a couple of fingers between them, but I have two suggestions that may be less invasive that cutting/welding:
1. I replaced my stock seats with '87-'93 fox body seats, more for safety and comfort than height. The '67 seats stopped at my shoulder blades so I was always afraid of whiplash should I be hit from behind and my back would get sore on long rides. The foxbody seats will recline so a little tilt gives you more head room as you go back. I used the foxbody sliders and chiseled the 'feet' off so that they sat at the same seat height as my '67's. I used a 1" wideX1/4" thick piece of steel flat bar and some carriage bolts to mate the foxbody sliders to the seat risers. This also allowed me to mount the seat back a couple inches, now I don't even need full extension to get comfortable. The seats also have an electric lumbar and headrests. TMI makes seat covers for them that are stiched like the classics to match my back seat and look like the originals. I liked this option as I drive the car more since doing it, feel more retained in my seat and am also adding 3pt. belts with chrome buckles to appear '60's. I don't weld and used a drill, hacksaw and files to do the install.
2. Go UP, I was considering this also, replace the hanging headliner with a foam contour fit one like some newer cars. I spoke with a local upholsterer and they use a foam and then shave/file it to shape. I don't know how easy it would be to do DIY or how much $ it would be to have an upholsterer do it , you may want to get a couple quotes. That should give you an inch or two there, I've never had mine out to measure how far the stock type hangs.
Good luck,
Jon