Word for the state of being compelled

Solution 1:

From the question:

I could only think of "compellement" but I fear it is not a word, or the state of being compelled exists in a different form than what I can guess.

As was stated in a comment under the question, the noun is not compellment, but compulsion:

[Merriam-Webster]
1 a : an act of compelling
// tried to get them to cooperate without using compulsion
: the state of being compelled
// He was acting under compulsion.
1 b : a force that compels
2 : an irresistible persistent impulse to perform an act (such as excessive hand washing)
// her compulsion to repeatedly check and recheck the stove to be certain that it is turned off
also : the act itself
// Gambling is a compulsion with him.

So, the example sentence becomes:

As opposed to doing it out of addiction, he did it out of [a] compulsion for the art".

Solution 2:

If you mean being compelled to do something by some internal force, rather than an external force, then driven is as good as anything.

driven adjective

Someone who is driven is so determined to achieve something or be successful that all of their behaviour is directed towards this aim:

Like most of the lawyers that I know, Rachel is driven.

Cambridge Dictionary

Solution 3:

An option would be dedication:

the willingness to give a lot of time and energy to something because it is important

An additional example (from the source):

She applied herself to the task with enormous energy and dedication.