Do you have more detail on how collection messes with MLAT calculations? It seems like for triangulation purposes, each device should produce some precise timestamp that is not affected by store-and-forward transmission.
I don't think the programs are made to handle that use case. At present you register your receiver's location with the server once and send packets as they arrive, that server doesn't know that your packets are from multiple receivers at multiple locations.
Unfortunately, there is no good solution for this. Either you rely on the time settings of the sender or on the time needed to transmit the data and on the time the data gets transmitted.
If we assume the sender is not malicious, a properly configured NTP client ("the time settings of the sender") would probably suffice. If the sender is malicious, unless you somehow make the "signal + timestamp" from the receiver non-modifiable from the receiver to the service, you've lost anyway.