Hello   @Regulus 

Updating firmware itself does not require  Internet connection on the range extender.

1. To update firmware though wired connection,   you can download firmware to computer first , then connect PC-- range extender via Ethernet ( PC has no internet) and login web page to update firmware.

2. It is also fine to update firmware through wireless but make sure you did not accidentally disconnect wifi during the update process. 

As wired is more stable, it is more recommended.

For RE450 OneMesh feature, only RE450 V4 and above version supports Onemesh, you may also consider another model if there isn't V4 found in the local store. 

https://www.tp-link.com/onemesh/product-list/#range-extenders/product-list/#range-extenders

 Hello  @Regulus

Yes, we can update firmware for range extender when RE & PC has internet as well. 

For example:  Router 192.168.1.1 <Wireless>Range Extender 192.168.1.X <Ethernet> PC 192.168.1.5

Range extender will receive an IP address from router, you can find range extender's IP address on router DHCP client list, and use that IP address to login web page of extender and upgrade firmware for it. 

For some new models of range extender, you can also do an online upgrade through wireless for range extender on the Tp-Link Tether app.

Hello   @Regulus 

Glad to know you solve the problem.

After you set up the range extender, router will allocate a new IP address for the range extender, and that new IP address can be found on router DHCP client list

It would be easier to use the new IP address of range extender to login its web page. If your router is on 192.168.1.X subnet, it will assist an IP address 192.168.1.X for range extender 

here is an example:FAQ: How to use my Range Extender properly if your range extender is configured correctly?