The question is, do you posess the skills to do the job to get the neck in the same shape as the one you played? Because if not, you're going to end up with a ruined guitar. If you do have your mind set on this, I highly suggest searching the Austin pawn shops and picking up one of the many import LP copies that where there when I went to UT a decade ago and spent all my time in pawn shops looking for guitars. Practice on one of those before ruining your Epiphone. Or better yet, go pick up a length of maple at least 3"x3"x12" and try shaping that.
Jimmy Page LP: $$$$$
Replacing LP neck: $$$
Reprofiling the neck yourself:
When I was young like you, I did some pretty stupid things to my most prized guitar. Now I'm paying almost more than what I bought it for to put it back together after 12 years of it being in non-playable shape.
Sure. If it were me, and I didn't have access to the neck and a copy carver, here's what I'd do:
Get yourself a contour gauge. You can get them at Home Depot, Lowes, Ace Hardware, Harbour Freight, Sears, or just about anyplace that sells tools. The large home home improvement stores generally keep them in the power tools section. They're normally used to copy molding controus, curves, uneven edges, etc. Make sure the pins/teeth are at least 3" or more so you can get a complete measurement from back of the neck to the top of fretboard.
Go back to the music shop and ask them to let you take some measurements of the neck. Take contour measurements every 1-2" and trace them on paper, labelling them in order from body to headstock or headstock to body. It's probably best to take a measurement on the neck for each contour trace so you know how far up/down it you were when you took the measurement.
After you've done the whole neck, thank the music shop for letting you do something crazy in their store , and take the copies home to transfer the neck profiles to poster board and cut them out as templates. you'll want to cut them out as a reverse of the trace, meaning they should look like the contour gauge when you applied it to the neck, or more directly so that the part that the neck would fill is cut out on the powerboard.
Use them to gauge how much you need to shave off your guitar neck by putting them at the same locations. Or, you could do the same thing you did with the Jimmy Page and take measurements of your neck at the same locations and compare the two to see how much work is going to need to be done. You might stop right there.
Then get sanding. Even with an aggressive sandpaper, it'll take lots of time. Be sure to wear a good dust mask, even though you'll be working outside (you will be working outside!). Your lungs are very hard to replace. I personally use a respirator with particle filter attachments from 3M instead of a regular dust mask. Go slow, sanding only by hand, and check the contours often with your powerboard guides. Go
And I've never done what you're asking, but that's just the process I would follow to make sure I didn't screw it up. If you had access to the Jimmy Page LP and the ability to put it in a copy carver with your guitar, it'd be a lot easier. But my advice is don't do it, and don't hold me responsible if you mess up.
Remember the Alamo, and God Bless Texas...