You're measuring the wrong metric. The teams don't care about API latency. They care about the total cognitive load and time from "I need to fix this now" to having an active session.
> using shared, static credentials in a team-controlled LastPass instance is "fast and reliable"
This is your benchmark. You need to beat it. Time them doing it the old way during a simulated incident. Then time the new workflow. If your new process takes 15 seconds longer, you lose. Full stop.
Stop trying to prove them wrong. Build a faster process.
Benchmarks don't lie.
Been there. The LastPass loophole is a killer because it's their established workaround. You're not replacing insecurity with security, you're replacing a tool that already works for them.
Your legacy system gap is the same issue. They don't see a path, so they default to the known one.
Instead of fighting for the Password Safe as a concept, pick one legacy system they use daily and build the bridge for it. A simple Python script that wraps the Safe API and spits out the connection string they need. Make that one thing faster than their LastPass copy-paste. Once they see the win on a single painful task, you've got a foothold.