A word/phrase for “ not wanting to draw attention”

[Wanting] to keep a low profile is a common expression for this.

keep a low profile

• to avoid attracting attention to yourself:

He's been in a little trouble recently so he's trying to keep a low profile.

[Cambridge Dictionary]

There is also to keep your head down, which carries the sense of avoiding (a real threat of) trouble:

keep your head down

• to try to avoid trouble or involvement in a difficult or dangerous situation by behaving in a quiet way, so that people will not notice you

After unity, he had little time for Christian Democrats who had kept their heads down under the old regime.

[Collins CoBuild Dictionary]

Any clandestine approach to an operation could be said to be

backstairs ('backdoor' usually implies dodginess)

backstairs [adjective]

• involving intrigue or scandal; secret

[Collins

or a sub-rosa activity

sub-rosa [adjective]

• secretive, private

sub rosa [adverb]

• in confidence: secretly

[Merriam-Webster]

And there is always the very safe option low-key:

low-key [adjective]

• not elaborate, showy, or intensive; modest or restrained. [discreet]

their wedding was a very quiet, low-key affair

[Lexico]

I also want to say "Inconspicuous" but for some reason @desbest's answer was downvoted. I will explain my answer here. According to the Cambridge dictionary, Inconspicuous, adjective, means:

not easily or quickly noticed or seen, or not attracting attention:

So, you can say "He tried to be as inconspicuous as possible"

You can also say stealthily (Cambridge dictionary):

quietly and carefully in order not to be seen or heard:

"He moved around the building stealthily"

You can also have the adverb form of Inconspicuous, "inconspicuously" and the adjective form of stealthily, "stealthy".

That behaviour is described by this sense of discreet:

[Merriam-Webster]
3 : UNOBTRUSIVE, UNNOTICEABLE
// followed at a discreet distance

In other words, you have to approach somebody in order to get something changed (you're not being clandestine, where nobody else knows), but you only want the person you're approaching to know about the problem, and not anybody else.

You get something done, but not in a way that causes any kind of public or political issues, or even any awareness.

I talked to a particular person in private, because I wanted to be discreet.

Your question does not make a moral judgement about why it is desireable to avoid being noticed. I offer three possibilities, in order of most negative to most positive connotations ("furtive" suggests a thief, but "stealth" can have an arguably good purpose in a war effort, for example):

Furtive (of people) behaving secretly and often dishonestly, or (of actions) done secretly and often dishonestly: "I saw him cast a furtive glance at the woman at the table to his right."

Surreptitious done secretly, without anyone seeing or knowing: "She seemed to be listening to what I was saying, but I couldn't help noticing her surreptitious glances at the clock."

Stealthy quiet and careful in order not to be seen or heard
Also Stealth: movement that is quiet and careful in order to avoid notice, or secret or indirect action

(Cambridge.org)