Re: What is the best approach to overwrite template tags?
hello Joe
As far as overwriting the template tags are concerned,That is not allowed.You can only overwrite pluggable functions(In case of wordpress/buddypress).
As far as I see, the bp_login_bar() is a normal function(template tag), you may have a look at the code in buddypress/bp-core/bp-core-templatetags.php,so you can not overwrite it.
Yes, if you want a customize the look of the form/output generated by it, you can use a work around.
Copy the code for bp_login_bar and put it inside a new function say bp_custom_login_bar() inside your theme’s functions.php,now you may modify the code inside the bp_custom_login_bar() to make it look as you want.
The only extra work you will have to do is, you will have to replace all the calls of bp_login_bar tag in your template by bp_custom_login_bar.
Hope it helps.
If I misinterpreted your question,please clarify.