If they require you to use the bastion, then trying to avoid it is probably a bad idea.
If the bastion is running an ssh server, you can jump through it with ssh pass through (using -J).
SSM provides session manager which allows you to skip having a bastion altogether- it basically lets you start an “ssh” session to a private instance without opening ports or networking using aws creds. This requires that you have access permissions to do this and that ssm is enabled.
But… if the reason you are using the bastion is so that they can inspect the traffic, then they’re not gonna let you bypass it via ssm because that also bypasses the managed networking.