You use the phrase "gain autonomy" in your question and the verb autonomize does exist. It's fairly uncommon but it should be readily understood.
Very few dictionaries carry autonomize in either -ize or -ise forms - a OneLook search finds entries only in Merriam-Webster and Wordnik.
Merriam-Webster gives the following definition and examples:
autonomize transitive verb
variants: also British autonomise
to make (something) autonomous
"over time some systems of rules become autonomized as legal systems."
Michael Kaern et al., Georg Simmell and Contemporary Sociology, 1990
"the present regime has deregulated and autonomized the financial
institutions which is a hallmark towards economic revival."
Pakistan Newswire, 10 Apr. 2002
Autonomize does have nearly the meaning you want, although it should be noted that rather than meaning "become independent" its meaning is closer to "grant independence", so it is broader.
Also, it wouldn't really work in your example sentence, being of the wrong register - it's a technical and formal word, one often used in academic political and economic contexts, as the following examples found with a Google Books search show:
"Since 2003, however, the tendency to autonomize from NATO and US influence has been translated into open inter-organizational rivalry in the operational, institutional and identity domains."
The Influence of International Institutions on the EU: When Multilateralism hits Brussels, O. Costa, K. Jørgensen (2012)
"Neo-liberal economic growth encourages autonomized regions. The autonomized region is the EU's accommodation to a combination of globalization and neo-liberalism."
The European Union: A Political Sociology, Chris Rumford (2008)
"To insulate and to autonomize the judiciary under these circumstances accomplishes nothing but to insulate and autonomize corruption."
Judicial Independence in Transition, Anja Seibert-Fohr (2012)