I'm sorry your Cthulhu is sick.
Unfortunately, what you have described is very much part of pet ownership. When our little ones are sick, we must follow the vet's, in this case the avian vet's orders and follow the treatment protocol.
This goes for all species of pets, even though they may not like it, the fact is that it's for their own good.
Depending on the budgie's personality and the type of bond shared between budgie and owner, some budgies can be more accepting of the medicine via beak when compared to others. The behaviour in some can even improve as they realize what we are doing is actually helping them out so that they get better.
I can give you an example of how I have bonded with one of my budgies: My Tito was a lost stray young budgie who came to me on the sequence of a dog attack. My neighbour found him on her backyard sitting on a laundry's basket when her dog saw the bird first and went for the attack.
I had no previous bond with Tito, at that time I didn't even know if he was going to make it. He was a great trooper during the treatment process, I had to clean his wound on a daily basis and whenever needed as well as giving him medication to prevent infection.
He was accepting of all of this. During the treatment, I would talk to him very soothingly. In less than one month, he was all healed up and to my surprise he even picked some some of my comforting words I said to him during treatment. We have bonded deeply out of an unfortunate accident and for the following years I have had him, Tito made sure to always show the love and gratefulness.
In order to minimize Cthulhu's stress levels, you can have some soothing music on low volume for him, cover the cage on 3 sides and dim the light when you are about to catch him as gently as possible.
When administering the medication, you can try to calm him down by talking to him in a reassuring, comforting way and praise him after each dosage of treatment as he goes back into the cage. You can also offer him a little tasty treat for him to eat after getting his medicine.
During this process, it's important to try to hide your fears/anxiety and any negative feelings and to make use of your confidence as a pet bird owner and your determination in doing your very best to heal your budgie, having the needed bravery to face adversity.
I'm sending my best wishes for Cthulhu's steady and full recovery.