RBR750/RBS350 won't sync and have no way to update firmware

My guess is the original packaging is long gone.  Netgear prints the "kit" number on the box, as in RBK753 ("K"
 for "kit")  Netgear packaged all sorts of Orbi products into various "Kits".

It would help to report the specific model numbers printed on the product label on the bottom of each unit for the router and two satellites.  That will clear up any confusion over which firmware file to use:

• The RBS350 satellite indeed has a firmware file ending in *.img, whereas
• The RBS750 satellite has a firmware file ending in *.chk.

Firmware updates on Orbi systems are indeed awkward.  When the actual firmware does not match the latest version shown on the support web site, customers have the option of downloading the firmware, unpacking the zip file, and using the "Manual" tab on the Firmware Update page to load the firmware.

Obviously, when the router reports a satellite as "Out of Sync", it has no reliable connection to the satellite and updating the firmware is going to be a challenge.

It might be worth trying this:

• Power off both satellites.
• On the Attached Devices web page, both satellites should show as "Disconnected" and there should be an option to "Delete" them.  Do so.
• Select one of the satellites and do a Factory Reset.  i.e. hold a paperclip in that tiny Reset hole when you power on the satellite and hold it until the power LED begins to flash Amber. Let it boot up.
• Use the web interface Basic Tab menu item, "Add Orbi Satellite" (bottom item) to add that one satellite while the satellite is in the same room with the router. (makes watching things a lot easier).