Are you using Bitbucket Cloud or Bitbucket Server?
Bitbucket Cloud (bitbucket.org) has a premium feature that lets you whitelist IPs: https://confluence.atlassian.com/bitbucket/control-access-to-your-private-content-862621261.html
Bitbucket Server (the on-premise version) is installed within your network anyway, so your team should be able to restrict access accordingly.
That's a job for your network team.
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