Truth be told, I doubt it will be much of a battle at all. The government will regulate Bitcoin mining -requiring miners to only confirm transactions from approved exchanges. The regulation will appear reasonable and will get compliance from large mining pools (majority of the hash rate) and support from institutions involved in Bitcoin. Many private Bitcoiners will not want regulated Bitcoin, but they will be a minority. The majority of the hash rate controls the entire blockchain. Ultimately the most effective choice for independent minded Bitcoiners will be to create their own much smaller fork of the Bitcoin Blockchain.
The U.S. government certainly has the competence, resources and desire to regulate Bitcoin. Regulation of business, banking and money is a government specialty. The government regulates frequently and very few resources are required. The cost of enforcement can even be paid for with the proceeds of seized assets (unregulated coins). The Biden adminstration has just informed the world that the U.S. government does have the desire and intent to regulate all cryptocurrencies. To control all Bitcoin transactions globally, all the government needs to do is regulate a majority of the hash rate.