Setting up Zigbee network on 4 floors: ZHA or HA supported Zigbee GW?

That does not meet your requirement of different brands as IKEA Gateway only support IKEA devices.

Suggest instead that you just buy → ITead's "Sonoff Zigbee 3.0 USB Dongle Plus" (model "ZBDongle-P") based on Texas Instruments CC2652P +20dBm radio SoC/MCU

Tip is that you could even buy several of those as they are already great as Zigbee Router if flashed with different firmware and uses in with a standard USB power-supply. They are just not as as nice looking as the IKEA Trådfri Signal Repeater product. Flashing firmware is done relativly easy over USB port.

That is, start out with cheap USB adapters that can later be repurposed as a Zigbee Router once you have decided if you want to go with ZHA integration or Zigbee2MQTT, or both which is also an option.