Offer solutions, not suggestions
I’ve played and DM’d online enough to be on either side of this issue. Please don’t assume your DM suddenly becomes a stubborn jerk when you mention VOIP.
When you “constantly offer to switch to a VOIP client,” your DM may be hearing it as complaints, as requests for an (immediate) fix.
It’s easy enough for players to log into VOIP, and then the web system, and control their single PC.
Your DM is already juggling a fair deal of technology when you play online, controlling the monsters/ NPC’s, tracking Hit Points, communicating with all the players, switching maps. That can occupy a DM’s full attention, espcially for a table like yours, where you only play online occasionally.
Also, realize chunk of your DM’s free time gets spent preparing your next adventure (especially for online gaming) and not getting comfortable with your favorite VOIP technology. (If he’s less familiar with it, he will probably be less confident it will really improve things.)
Congratulations, you are promoted to Communications Officer
Since this issue is bothering you, take personal ownership of the communications issues and solve the problem for the table.
The next time you are playing in-person, show up early and make sure your DM has everything set up on their system to use your favorite VOIP tool easily. Let your DM know you want to take charge of troubleshooting any communications issues for the group, so he can focus on running the game.
You could DM a one-off evening yourself as a dry-run, but that’s not necessary. Just having a time set aside to for everyone to log into both systems and get everyone connected.
The key here is that this is done at a time when the DM is not expected to run the game. Nor is he expected to help get anyone else’s system set up.
Then, the next time you play, get everyone logged into VOIP before play begins. You provide any necessary troubleshooting for anyone having trouble.
Let me know if your DM still refuses.
Heck yes, the DM deserves to be treated differently
Comments on this question that the DM deserves no special consideration here because it’s not a rule issue are misguided. DM’s spends their free time preparing adventures, effectively providing free entertainment to their players. You can always find a new player; sometimes you have to hire a DM.
Handling an issue like this for your DM is a great way to give back a little.