For long-term orthopedic and functional success, you should be able to pick up a heavy object, stabilize, then walk with it. The inability to do so is a sign your grip is fragile and that you're susceptible to chronic issues in places like the lower back, shoulders, and elbows. So don't neglect these.
Not many gyms have dumbbells that go up into the 200's, so don't think carries are limited to dumbbells. If you want to train grip with continuous progressive overload, the trap bar or farmers-carry handles, which can be loaded with weight plates, are your best bet.
How Heavy Should I Go?
Test your grip strength. If you're using a trap bar, get two times your bodyweight on the bar, deadlift it up, and walk with it in a slow and controlled manner for 20 seconds. Can't do that? You have some work to do.
Use a variety of timed sets and distance-based sets. Challenge yourself and stay fresh by altering training parameters. You can probably do more than you think. If the load doesn't scare you a little before you pick it up, you aren't going heavy enough.