If you don’t have a 78 turntable, it may not be worth the trouble in my opinion. However, if you knew what key they should be playing in on the songs, you could gauge that post processing.
Anything with pitch control would probably work. But I really don’t know if you’ll ever get it right recording it into your DAW at the wrong speed. You could always look for an antique shop and see if they may have a cheap turntable that does 78. That’s your best bet honestly. You’ll have to see if it’s worth whatever the purchase price is.
Or, you have the client spring for a turntable capable of doing this so you’re not stuck with it. Does anyone you know have one? Maybe you can work out a deal where you rent one from an antique dealer? Tell him you just need it for a few hours. Worst case scenario, pay a rental fee and buy a new needle for it. I wish you lived closer as I have one here I believe that goes from speed 16 to 78. It’s part of an old Sanyo stereo system I have that I bought in 1985 with all the rack components. If you really want to take this job on, it would probably be easier to have the 78 turntable capability.
All of the above said, you can probably figure out the math to this with a little work. The issue there is having the right tools in your DAW. I know the old cool edit or last version of Adobe Audition could handle it. I have that here and you can numerically type in the speed of your choice. What I don’t know is how speeding up the wrong speed will sound. You may not be able to get the right sound once you speed it up due to the speed being wrong. Ever hear a sampled sound once it gets out of the sample range it was recorded in? You may end up with that artifact. This is definitely a challenge. Whew…good luck!