We are in Australia and have recently (last 8months) changed to a dojo that sounds much like your new dojo. eg; bowing, respect, stances answering uss sensei when spoken to.
We wear our old ghi's (plain white, no markings) to the new dojo and use all our old gear, not dojo branded. We would not wear anything with markings from another dojo to our new dojo.
While I do not believe that our new dojo would have a problem with it, I do think that it would set us apart from the other students/families and may come accross as us not being ready to fit in and support the new dojo.
The gear bag I can understand, but actual clothing that you wear around, its like branding yourself the property of another dojo. Yes, many people cross train between dojo's (we used to) but most do not advertise it.
One thing our dojo really pushes is fellow student support. If this is a dojo you want to be involved with long term, then you need to show your support and let the other students know your not just passing through.