Difficulty with name servers when changing hosts

My domain was transferred back to me from my previous web developer three weeks ago, and is now registered with Cloudflare. I am in the process of moving my site hosting back to Bluehost, where it had been for 10 years prior. The domain can’t be transferred for another 5 weeks due to the 60-day lock after transfers, and since the domain is not registered with Bluehost they do not allow me to make changes to the name servers they provided on their end, so I am unable to enter my Cloudflare name servers. Through Bluehost Help posts I’ve found that;

If you want to host your website or manage your domain at Bluehost you need to use our name servers. The company you bought your domain from should give you the ability to change your name servers.

Is the only possible way to do this is to have to pay 200$/month for the business plan in order to be able to generate custom name servers, just so I can make those name servers match my Bluehost name servers (which kind of defeats the purpose of using Cloudflare in the first place)? To pay 400$ only for the ability to connect my domain to my hosting provider (which I already started paying for but can’t connect to) for 5 weeks seems ridiculous, and staying with my current host provider for another 5 weeks until the 60-day lock on my domain runs out isn’t an option either for security reasons.

I’m new to this but I’ve been researching this for a while now and I feel like I’m running in circles. Any help on how to proceed connecting name servers of my Cloudflare domain/account with my Bluehost web hosting would be greatly appreciated.
Thanks in advance,

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