After choosing Settings -> Backup -> [ Download Backup File ] while in a Microsoft Edge InPrivate session, browser may not display own dialog box...

まず、Microsoft Edge の InPrivate モードで報告された Settings → Backup → Download Backup File のフローを再現します。可能であれば、指定された Edge と Windows 7 の環境を使用してください。標準の Edge セッションと動作を比較し、繰り返し行ったダウンロードを正しく開く、保存する、またはキャンセルできるかを確認します。拡張機能とブラウザーのどちらに不具合があるかを特定し、再現可能な結果を文書化できれば完了です。

Please note that I'm unsure if what I'm reporting here might be an 'Authenticator: 2FA Client' extension v6.3.3 bug or not. But anyway at least what I'm reporting here might just be useful for troubleshooting purposes.
N.B. All my tests were made using Microsoft Edge Enterprise v109.0.1518.140 (Official Build)(64-bit) which is last version ever for Windows 7 SP1 x64 platform.
After choosing Settings -> Backup -> [ Download Backup File ] while in a Microsoft Edge InPrivate session, (and while Authenticator icon was not visible in browser toolbar) browser may not display own dialog box with [ Open ] [ Save As ] buttons, so end user doing this operation for 1st time, may not really understand what's happening and may even repeat pressing [ Download Backup File ] more times without properly noticing that anyway at least browser download icon has properly appeared in browser toolbar. So he/she may decide to repeat same operation from within a standard Edge session, and also after enabling Authenticator icon display in browser toolbar, after choosing Settings -> Backup -> [ Download Backup File ] he/she'll be able to see expected browser own dialog box with [ Open ] [ Save As ] buttons, and by so doing he/she'll also be able to notice that all his previous attempts to [ Download Backup File ] were indeed properly queued and are still awaiting to those actions to be completed. Now if user should decide to cancel last attempt to [ Download Backup File ] and to revert to previous Edge InPrivate session, he/she'll also unfortunately be able to verify that only 1st queued file 'Save As' action will be completed correctly succeeding in saving file with 'authenticator.txt' proposed filename in chosen folder, while his/her additional attempts will only always fail. When trying to 'Save As' OS dialog box will only open with an empty filename and proposed file type maybe even be 'File.ME' instead of typically expected 'Text Document (*.txt)' especially if chosen folder name was terminating with an extension (i.e. ...\folder.me in my tests). And because any further attempt to choose 'Cancel' option at the very right of every queued file will also do nothing the only remaining option to complete actions for 'authenticator (1).txt' and 'authenticator (2).txt' still queued files will be to revert again to a standard Edge browser session where indeed both 'Save As' or 'Cancel' actions, if chosen, will then just complete correctly as normally expected.
P.S. Please also note that I'm unsure if above reported issue only faced when operating from within an Edge InPrivate tab session (equivalent to Chrome Incognito tab or a Firefox Private tab) might really depend from 'Authenticator: 2FA Client' extension v6.3.3 code being used. If not reproduced with latest and more updated Edge Enterprise versions chances are that issue might even just depend from browser code while handling add-on/extension operations from within an InPrivate session tab. JFYI I'll also report issue to Microsoft directly via Edge own 'Send feedback' feature, and if you too can reproduce same issue I described above, then I suggest you to please do the same.
P.P.S. My Windows 7 SP1 x64 Ultimate client was also quite memory loaded (a monitoring tool called 'Memory Defragmenter' shown as icon from SysTray area also typically showed memory usage of 89% during my tests even if I have 24GB RAM) but this may even not have any kind of dependency on results I experienced since a solution was always found while operating in an Edge standard session.
Regards