This question is more of a theoretical/opinion-based nature than technical, but let me provide some answers.
Will client-side rendering open up security problems?
Generally, web application security is a server-side concern, not client-side. You can do things like input validation on the client-side, but the minimum practice for security is to sanitize, validate, and authenticate all request data anyway, so the client-side checks are more of for convenience and improved user experience than security. I'm not saying that there are no such things as client-side security concerns, but it's something I don't think is generally a cause of worry. Client-side rendering specifically and especially doesn't sound like something to be careful about: regardless of what your client-side code does, whatever <form> and <input> markup it generates, your server-side code should always handle the submitted data as if it could be malicious.
Is client-side the better way to go?
There are so many more factors to consider in order to answer this, so it's largely a matter of opinion. But since you're asking about Django, then you might want to reduce overall development friction by maximizing Django's features and design—and Django, in my view, is largely a static markup-first framework, meaning minimal use (at first, at least) of client-side JavaScript. Django Forms and Class-Based Views (CBV), for example, work well together to allow rapid development of non-single-page applications.
Your specific use case of an initial drop-down choice determining the main form to be presented could be developed very rapidly in the traditional Django way by giving up your single-page-application requirement, and just providing some initial menu page that will lead to the different views and forms (pizza vs. drinks, etc.), the latter of which you could build rapidly with the help of CBVs. (By the way, your specific use case doesn't seem too unique, actually. It's just the fundamental issue of complexity for which we have programming concepts such as polymorphism and inheritance in object-oriented programming—hence the appropriateness of CBVs.)
I know that single-page applications are nice, and is the fashionable thing nowadays, but I think people underestimate the speed of old-fashioned HTML applications. And by speed I mean not just the user's client-side experience (HTML pages load rather rapidly with HTTP2 and CDNs and all the other modern Web infrastructure tech these days), but also development time.
Besides, you can always just add single-page-like experiences in a progressive manner. Django is particularly suited to an agile-style development strategy where you'd build initial functionality rapidly without much client-side JS, and then just add rich client-side experiences (using React or Vue or something similar) where it will add the most value for users.
I also want to minimize the amount of code that a competitor can 're-use' from my work.
I don't know the full context, but generally I wouldn't worry about this. If you won't do much client-side rendering, then there won't be much client-side code to ‘steal’. But even if you do, unless you specifically write your client-side code in a way that maximizes reusability (either for yourself or for others), I think coders tend anyway to write highly-coupled code, which is to say, your client-side code will tend to be highly dependent on your server-side code's specifics, which means poor reusability. Your competitors could copy your client-side code all they want, but the cost of making it work with their own back-end will be so high that it wouldn't be worth it, they'll just want to write their own.
