I had an ML roster player who I designated with waivers.
He cleared.
Now if I try to assign him to AAA, it says:
[Player] may only be assigned to the big league team due to his current contract and option status.
If I try to assign him to ML, it says:
[Player] may not be assigned back to the big league team since he was designated for assignment. You may trade him, release him, or assign him to a minor league club if he clears waivers and accepts the demotion.

Is there away around this besides trade or release?

If he cleared waivers, you should be able to put him wherever you want. Open his player card and make sure his waiver status says Cleared.

I suspect the player has more than five years MLB service time, so he can refuse assignment to the minors. Far as I understand it he is in limbo, and you have to trade or release.

TBH the FAQ could use a rewrite on how to use DFA and waivers.

Definitely cleared waivers.
and says this when trying to do it again:
[Player] is either on the waiver wire, has already failed to clear waivers, or has already cleared waivers. There is no point to waive him again.