I have two boys who are autistic, they were very similar to that at age three.
Firstly it's not a temper tantrum/trying to get his own way, autistic children get very caught up in obsessions and routines. The DVDs likely help him feel safe. My younger son (age 6) is currently fixated on one single episode of Minnie Mouse Club House which features Minnie Mouse opening a bow shop. He shouts 'Minnie Mouse' from the moment he gets home from school and watches it on repeat. It helps him feel calm and settled. Prior to Minnie Mouse it was The The Night Garden, the episode finished when we were out the room and he put his head through the TV as there was no one there to put the next one on immediately. My older son (age 9) will watch programmes on the iPad. He chooses a small section of dialogue usually only a few seconds long and watches it over and over again.
I let them get on with this because it seems to be what they need. I encourage other types of play and they tend to seek other things when they feel more chilled but that play is still very routine and repetitive.
I understand that it's not possible for you to be as flexible as I can be on screen time. A visual timetable and a strong routine would be the best help in teaching him when he can watch DVDs. Visual timers like sand timers are really helpful in showing when DVD time is coming to an end.
Other toys I have found successful with my two include anything sensory, we have had magic sand, lentils on a tuff spot with things to scoop and pour, Play Doh, slime, disco lights, liquid timers, very soft blankets and glitter wands.
Both children have responded well with having somewhere to calm down, kids play tents are great for this and are fairly cheap. You can get dark dens and glow in the dark toys but basic play tents work fine. Putting in soft pillows and blankets really helps.
My youngest also loves watching things move, like a train set where the train moves round the track on it's own and he can watch, he also loves things that spin. My older son has always loved toys that you can get it to repeated the same phrases over and over again (you probably won't like this though!). Things like the vTech toy phones and Leapfrog laptops.
It might be worth using some Makaton with him, even if he won't look at you and doesn't seem to be playing attention. Something Special is brilliant for picking it up and he might enjoy it.
With regard to the suggestion that this little boy should be placed somewhere more specialised there is nowhere. That's not to suggest that you are obligated to continue to care for him if you feel this isn't working out, you aren't. You have other children and your own needs to consider. I just come up against this a lot so wanted to address it.
My children are considered severely autistic and were both diagnosed at age two, my oldest child attended a special needs preschool for half days as that is what was available. My youngest son attended the same special needs pre school but by then the funding was cut so he did half days and only a partial week. Now funding has been cut even more and that preschool is gone, there is another preschool with a special needs speciality but I believe they are either closing or closed. There is nothing else. My oldest child had Portage for about a year, my youngest child was allocated six weeks, my neighbour's young child who is believed to be autistic can't get Portage at all. My older son got speech and language therapy at home until he went to nursery and they took over (in house speech and language therapist), my youngest son had to wait till nursery. My neighbour's son probably won't get to the top of the waiting list before he starts school. There is no help, there is no support, there is literally nothing.
That said if I can be helpful in a very non expert and not based on my personal experience way please feel free to message me I will do my best.