Not 100% sure, but isn’t the GitLab integration bound to the user account? Not sure if a webhook has enough permissions to access the other integrations anyways, you might be in need for the use of a personal access token for your account, not sure if that makes a difference, just throwing out ideas here.
I’ve never worked with Zapier, but from what I read they should also be able to run more complex queries for the destination system, so if you give them your personal access token and authenticate with it on the ExecuteCommand API call, you should be able to directly trigger the slash command in your name by skipping the webhook and the slash command parsing entirely.