Not sure that you understand the severity of the issue of casually 'temporarily using Cloudflare'.
Not using JavaScript or not keeping logs is pointless, if users' traffic, i.e. plaintext transaction information is literally decrypted and cached on Cloudflare servers. It threatens the whole concept of your service as a mixing site.
As I have been able to read on their website, they don't do the mix process on their server, a point in favor, since these data will not be available to any third party.
Well, there are no mixers that don't use Cloudflare or a third-party service like Incapsula, Cloudbric or DDOS-GUARD... In other words, any provider has access to the data that you mention.
Many mixers, "almost all" have been using cloudflare for years, and the others use third-party services such as DDOS-GUARD, which amounts to the same thing, since they perform the same function. In that case there would be no solution for what you propose.
As for Cloudflare, it has strict policies on the privacy of its users. In general, Cloudflare affirms that it respects the privacy of user data and does not sell or share personal information with third parties, and has demonstrated this on many occasions, including in media cases.
However, it is important to note that no online service can guarantee absolute data security, and this applies to cloudflare, Incapsula as well as Cloudbric or DDOS-GUARD.
All of them must follow the applicable laws and regulations in the jurisdictions where they operate. In this case, the user is the one who decides what to do and what extra measures to use to use these services, which are all the same in my opinion.