I'm not familiar with the internals of BitBucket, but I suggest you perform some user actions which cause pull and push activity, then under the Diagnostic Tools menu, use the CPU Profiler to explore CPU activity to see if you can determine a method that is invoked as part of push and/or pull operations. Once you've determined a Class.method, create a custom service for that method.
1st thing i did was to enable jmx in bitbucket
and followed this process
And using jconsole profiler i could get the class and method , which i have defined as "define java services" under custom service detection
Now i see the pull/push requests separate under the new custom service.