Yes, it is ill-formed (what a nice word! :) ) language. Only a noun or an adjective can come after become—because the meaning of this word is about coming to be a certain way. So, you should treat "become" functionally in the same way as you treat the verb "to be" in its simplest sense of, well, being something (the verb "to be" is usually followed by a noun or adjective). For example, one can "become worried" ("worried" is an adjective)—because you can say, "I am worried",—but not "become to worry" or "become to not worry". Technically speaking, one could say "I am not to worry", but here, the verb "to be" becomes a modal verb—this basically says, "I should not worry"; that is why I'm saying you could find equivalency only to the simplest sense of "being" in the verb "to be."
To address your specific sentence, the correct way to say it would be,
I'll definitely try it when I stop worrying about that.
If you need a sense that is not negative (i. e. you start to do something, not start to NOT do something, which can be transformed into "stop doing something"), you can say,
Once I have tried to do it my way, I will also try the method that you are suggesting.
This construction with "once + Present Perfect" can also be applied to your specific phrase,
I'll definitely try it once I've stopped worrying about that.
P. S. And yes, googling a phrase in quotation marks is often a good first step for determining whether native speakers use this phrase at all.