I have had the S20 Ultra 5g for over a year now and since day one, regardless the messaging app, WiFi drops when: sending and receiving photos, sending and receiving group messsages. Does not matter the wifi connection. MY trucks stereo, for example, connects via wifi for Android Auto to play (and I usually use Waze and Spotify. IF I get a picture message or a group message, wifi disconnects and I lose waze and spoitify and have to reconnect. If I am home, same things, if I am ANYWHERE, same thing. Anyway to fix this? It is not a router problem, because it happens on ALL WiFi connections. It has to be handset issue and resetting the phone has not fixed the problem. It's more a pain in the butt, if anything, especially when I am driving and need Maps. I was using the Verizon messaging app and it wouldn't let me send photos at all, and was having the same issue with Wifi, I use Textra now and it does what I need it to, and still drops wifi.
try submit a ticket through Samsung Members app which can be downloaded from Galaxy Store. This app will automatically collect the log about your wifi settings and agents will be able to tell you what the issues are about.
include your wifi modem, and provider so they can look into it more detail.
Its all wifi connections, all of them. Meaning, at home, when I connect to my vehicle, at work, at Starbucks, at a hotel, you name it, if I connect via wifi it will drop when MMS is sent or received. I will submit a ticket as you suggested, and give my modem and internet provider, but it isn't just when I am at home.
I would test the phone in Safe Mode to see if the WiFi drops when sending a message then.
The next step would be to do a complete backup and reset of the phone. https://www.samsung.com/us/support/answer/ANS00062029/
If you are still having Wi-Fi connection issues service would be recommended. Please send a private message with the full model code and IMEI number of the phone. http://bit.ly/2XERsky