After playing around with the Beach Boys 10" a bit, I suspect Capitol may have cut this in odd way (shocking, I know).
What I suspect they did was cut the record on the lathe at 45 RPM while playing the tape back slowly. Since Capitol hasn't been cutting 78s for a half century or more, it would not surprise me if their lathe can't spin at 78 RPM (I'm trying to find out whether that is the case or not). But no problem right? Play the tape (or digital file) back at around 57.7% speed, and spin the lathe at 45 RPM and everything should work, right? Not exactly. The problem comes when introducing the RIAA compensation curve, and if that is applied as the tape runs at the slower speed, it also shifts the RIAA crossover points. If that is the case, and the RIAA curve is off, the frequency response will be a bit off when actually played at 78 RPM.
How do I know this you ask? Well, I don't, it's a somewhat educated guess. I recorded my copy while playing at 45. Then I sped it up the proper amount using the method outlined in the post below. But when I used the EQ program I suggested (using the 45 > 78 setting) the frequency response sounded off (too bass heavy and muffled in the treble). But if I didn't apply the EQ change, it sounded right. When I recorded at 33.3 and applied the appropriate speed changes and EQ correction in Equalizer, the resulting file sounded even more off than when I recorded at 45 and applied the changes. I've used Equalizer to adjust EQ under similar circumstances dozens of times in the past, and this is the first time I've found the results unsatisfactory.
It was also interesting to me that on the EQ corrected copy I could see the brickwall frequency cut off appears to have been mostly shifted down to around 17 kHz, whereas without the correction I see the brickwall cutoff around 22 kHz (which is right were I would expect it to be if they used a 44.1 kHz digital source, or had a 44.1 digital delay head in their cutting set up). Also when I inspected the re-EQ'd file I recorded at 33.3 RPM, the brickwall filter appeared to have mostly been shifted down even further than that (to around 15 kHz).
So my revised advise--in this case only--is to skip the EQ step I outlined below for best sound quality. Just record at 45, speed it up, and don't worry about correcting the EQ. It will sound fine that way. The difference between the two is not so huge anyway, but the uncorrected version sounded more right to my ears. Or you can do it both ways (it's a free program and works quickly) and decide which way it sounds best for yourself.
I just want to emphasize that in other cases, I have found the Equalizer program to work perfectly for transcribing vintage 78s. Brian Davies, the author of the program knows 78s inside out and backward in his sleep. I highly recommend Equalizer in general both for transcribing vintage 78s at 45 RPM, or for recording 45 RPM singles at 33.3 (which is sometimes a useful technique if you have a noisy 45, but that's another story). And as I have noted in the past, Click Repair (also by Davies), is the single best vinyl restoration program I have used, and I recommend it highly to anyone who wants to digitize any kind of record (78s, 45s, LPs, flexi-discs, whatever...).
All of this is assuming Capitol used the RIAA curve to cut the record, and since the release doesn't indicate any other curve was utilized (Decca, Columbia, Westrex, etc.), I think it is reasonable to assume they used RIAA, although I could be wrong about that.
I have a few inquiries out, and I'll see what I can find out what actually happened. Although, honestly, I think I've wasted more than enough time on this already.
**Update: I revised this post for the sake of clarity. The original post was written quite quickly, and may not have been as clear as I hoped it would be.