How to express succinctly one who is not performing promises he made, but just trying to keep the issue (or you) under control

I did not realize that you wanted to describe his attitude toward you instead of his attitude toward his promise.

If that is the case, then you are being treated like a Red-Headed Stepchild:

A person or thing that is neglected, unwanted, or mistreated

or a second-class citizen:

A person belonging to a social or political group whose rights and opportunities are inferior to those of the dominant group in a society

Original answer below

The closest I can think of is procrastinating:

to be slow or late about doing something that should be done

This still implies a lack of desire but not necessarily because he made a false promise. The lack of desire may very well be due to higher priority issues as you noted, or may be due to some unpleasant aspect of keeping the promise.

I want to pay my bills, for example, but don't enjoy it so end up procrastinating

In your sample sentence:

You are procrastinating about keeping your promise.

Updated to reflect the change in question. In most situations in which someone is avoiding doing something, it is because they do not want to do it (which is where you get such answers as "slow-walking," "procrastinating," and "runaround").

If you believe that he does want to help, but other things continue to get in the way, you might want a way to say that he is not prioritizing you, or that he is not valuing you. In the form of a single verb, he is neglecting you, belittling you, denigrating you, or disregarding you.

Original answer: The term you are looking for is slow-walking.

https://en.m.wiktionary.org/wiki/slow-walk

Not keeping a promise is a breach of social contract. It is an act of 'bad faith'.

For example, Fred agrees (promise=contract) to go to the movies Monday night with you. On Sunday, an old friend calls and asks him to a baseball game. He'd rather go to the game with his old froend than see a movie with you. So, he phones you and cancels going to the movie with you (because he has found something better to do).

(Not worthy)

abdicate: fail to fulfil or undertake (a responsibility or duty). "the government was accused of abdicating its responsibility"

shirk: avoid or neglect (a duty or responsibility). "I do not shirk any responsibility in this matter"

Shirk is less formal and I'd say a bit more crude than abdicate. There may be other synonyms to those words that could be used like "evade" or "neglect", depending on the situation.

EDIT:

I thought of some more information that doesn't answer the question directly but could be useful culturally. If this information is being conveyed in an environment where politeness or co-operation is a priority, eg. a business, it's less common to use an accusation to describe an issue like this. This especially applies if the issue is being brought to the attention of someone for the first time or if it could be perceived that the person who made the promise is behaving reasonably.

An alternative is to describe the situation, and particularly the role the promise has in an outcome, and let the audience infer the accusation if they believe it is appropriate. An example would be "The report can be finalised once I get John's approval".