You could ask whether the client's Bugzilla is configured to accept tickets over email. See this Stackoverflow question for more details.
Another option would be to run something on the client side that can receive messages from you and then interact with Bugzilla. Alternatively, it might poll something at your office to receive Bugzilla requests.
You can install Greasemonkey (Firefox) and add a user script that removes the part of the webpage that is not relevant:
// @name Remove possible duplicates
// @namespace bugzilla
// @description Remove possible duplicates
// @include http://bugzilla/enter_bug.cgi?*
// @version 1
// @grant none
If you want to stop external access then one way is to set Apache Password Protected Directories with .htaccess File
This will mean any user accessing below a certain DIR level will require a password for access. Please note, this access is independent from the access granted by Bugzilla
A simple instruction guide is available here
This suggestion (and this seems to confirm this worked for someone else as well) to go into each product (in Administration) and uncheck 'Open for bug entry' to prevent entering new bugs, then (using the Edit Group Access Controls on that page) for each product set 'canedit' on for some group with no one in it (the explanation on that page says that there ...
I've used Basecamp and Jira separately, and Zapier seems to integrate both of those.
They also have a free subscription here
Fairly straightforward information here:
Integrate Basecamp with JIRA
If you find that your team is growing, or you need to integrate more issues you can always just subscribe to a higher tier.
+1 for Zapier