I like your thinking, but Im afraid its not going to help. I have just tried with a 030 card and have encountered the same issue. Also, Im doing this on a fresh Workbench 3.1 installation with no CGX installed.
As soon as an accelerator card is in place, the timing is broken. Then again, it could be broken without an accel card too, since the basic 020 is not able to play mods in realtime in that case and it may not have to shift as much as data in that case. Just guessing.
Are you making sure you are using the correct ahi.device and paula.audio for your cpu? Do you get problems when using the paula audio driver?
So ahi.device is for 020 through to 040
ahi.device.060 is for 060
Remember to manually rename cpu specific files around when swapping between an 020/030 cards and your 060 card - either that or re-install AHI with its installer and make sure it re-installs for correct cpu. I'm guessing this is how it works, but like I said, haven't installed AHI in 7 years.
Remember that the PreludeTapeDeck program doesn't need AHI, nor does play16 if you set the PRELUDE tooltype... so what happens when you use either of these programs? Does it work?