I spent some time on DAG based designs and this one reminds me of something that I rejected early.
As far I understand, the level of security against double-spends basically depends on the total amount spent on fees on the legit DAG. If so, an attacker just secretly builds a subDAG that includes a few double-spends, spends more on fees in the subDAG than was spent in the legit DAG, then publishes his subDAG. Now everybody should switch to the subDAG as it has burnt more fees. As long as the total amount of double-spends exceeds the fees, the attack succeeds.
Smart, sorry, I'd missed that from your original post. But it seems like V. Buterin's argument from the link in the OP still applies:
"This seems weak, but in reality it isn’t; we know that in the case of Bitcoin, once the currency supply stops increasing mining will rely solely on transaction fees, and the mechanics are exactly the same (since the amount that the network will spend on mining will roughly correspond to the total number of txfees being sent in); hence, fee-based TaPoS is in this regard at least as secure as fee-only PoW mining."
@TomHolden, I agree that Satoshi's PoW has the same potential vulnerability in that if double-spends exceed the value of what was burned to provide security, then a 51% lie-in-wait attack is possible funded by the value of the double-spends (possibly also shorting the exchange value in case the successful attack craters the price).
Thus, @tonych's concern applies to every consensus design (including Satoshi's) which is based on burning some resources as the metric of the longest-chain-rule (regardless whether multiple branches are merged to form the longest-chain, e.g. a DAG).
We really don't want a crypto-currency that is too liquid into fungible monetary equivalents, as it would not be secure except by overpaying for security. This points to future of crypto-tokens more for micropayments as a more viable reasonable security cost paradigm. As block rewards wind down for Bitcoin, then this may become more apparent.