Idiom Request – Is There an Idiom for a Strong Negotiation Position?

(Note that piece of painting isn't normally idiomatic in English.)

A very common idiomatic usage in this context is:

The police chief asked "Does this painting ring a bell?"

ring a bell (also ring any bells)
to sound familiar

Don't take that sound too literally though. The expression can just as well be used in contexts where it might be directly replaceable by ...seem, look, smell, taste, feel familiar.

In much the same vein there's also touch / strike a chord, which could (just about) be used in OP's context. But that one often has stronger allusions to strike / hit home which usually conveys particularly, often painfully relevant. As in His story about being mugged struck a chord with me (because I myself had been mugged, so listening to him brought back disturbing memories).

The phrase set for life was the first that came to my mind. It doesn't necessarily imply making a lot of money, but with some additional words you can make some idiomatic phrases.

If you make it big on Broadway, you'll be set for life.

You'd be set for life if you'd invested in the month after their IPO.

Another possibility is to strike it rich. This usually but not always means a person is set for life, but rather that they made a lot of money doing one thing or another.

By the Great Horn Spoon! uses this phrase a lot in the context of people migrating to mine gold. Example from the summary (likely many more in the book text):

They pick a beautiful spot, but then, as they hit bedrock, Jack and Praiseworthy find gold and strike it rich.

How these two relate to other answers suggestions:

• Anecdotally, make a pile is rarer than other suggestions here, even when spoken. The Ngram supports this notion, but make a pile's search results (18.5 million at time of writing on page 1) are greater than strike it rich's at 1.2 million. Then again, "make a pile" can refer to making a physical pile of objects, so it's hard to tell how many of these results are being used in the context this question is asking about, and paging to the end of either phrase's search results yields two more numbers (220 for "strike it rich" and 167 for "make a pile") which exclude similar results.

• make it big does usually refer to money, but "making it" can refer to a more general notion of success, with "big" simply amplifying it

• Here's sources for make a fortune versus make one's fortune. Making a fortune simply refer to making a lot of money. In context, this could be a smaller amount than needed to be set for life. On the other hand, "one's fortune" is all the money someone needs to be set for life.

• independently wealthy is a little tricky, and it is in with financial independence. It suggests something similar to set for life, that one's needs do not depend on someone else's wealth. However, the exact implications are different. Set for life seems grander and might suggest more money. Being independently wealthy or financially independent means that you do not have to work for money. See Reddit's r/financialindepence for a starting point on the connotations of these two phrases.

Ngram viewer with a few of these phrases

It's possible none of these phrases have the exact connotations and usage you're looking for. But if you combine them together and give enough context, you can build your own idiomatic phrases. For example,

He struck it rich selling carpets and was set for life by 35, but he kept working until 40.

What this says is:

1. He made a lot of money selling carpets.
2. He could have retired at 35.
3. He continued working despite not needing to, presumably for reasons other than money.