Oh wow good question! I dont think so bc Erlay‘s goal is scaling not privacy, as it assumes that „every unreachable node is directly connected to several reachable nodes, this policy ensures that a transaction is quickly propagated to be within one hop from most of the nodes in the network“ - I think that obfuscating one hop (sometimes, not all the time) as a byproduct would probably not be enough to reliably break BS heuristics, but I wonder if parts of the BIP could be used to improve the stempool idea - would def be interesting to see an analysis from a privacy perspective for this!
Doesn't erlay solve this problem by allowing a node to 'broadcast' transactions using reconciliation instead of flooding?
https://github.com/bitcoin/bips/blob/master/bip-0330.mediawiki
Discussion
I think you just need some way to prevent your tx from being flooded. If the default setting was to never 'upgrade' a reconciliated tx to a flooded tx it should diffuse slowly through the p2p network. Maybe this makes it more vulnerable to timing analysis but iirc there was some random delay built in. I'm fuzzy on details, it's been a long time since I studied the erlay proposal. Plus implementation details may differ significantly from the original paper.