My suggestions were shown below for comments. I will get back to you once it is confirmed by our senior and keep you updated the results. I will reply to you once it has been obtained the solution from our management and keep you updated in due course.

yes, but do mean "their" or "the" or "your". It depends a bit if you "own" the answer or are merely acting as a conduit between the client and the management I might try something like I need to bounce this past my management and I'll come back to you with answer as soon as possible (or set a time target).