Thanks @jsha. I made a concerted effort to reproduce the problems with Avast myself in a VM but was unable to. After following up with the client, I found that they are on Windows XP SP3, and their syptoms are consistent with the known problems above: trust problems in IE and Chrome, but working in recent Firefox.
Long story short: I think our problem was not Avast, but the fact that the client was on Windows XP SP3. In all the iterations I tested, Avast was working fine in Win7, and disabling Avast on the client's XP SP3 machine had (unsurprisingly) no effect.
So short of @mholt's anecdotes in the other thread, I've got nothing to suggest there's a problem with Avast, even when it's configured to MITM HTTPS connections. Sorry for the FUD.