How can I develop a pseudo-polynomial time algorithm for a non-integer problem?
Classifying algorithms as pseudo-polynomial time is not really a meaningful concept if there are no numbers (integers) in the instances. Of course, it is up to you to define what parts of the input should be considered "numbers".

Using real numbers in the inputs is a bad idea in the first place if you want to be able to analyze complexity or computability of the problem, as there is no way to encode all real numbers as finite strings from a finite alphabet.

For your problem, you may be able to turn it into an equivalent integer problem using some scaling, but it is impossible to say from the description given. Otherwise I would suggest you redefine the problem with rational parameters instead. This allows you to analyze it as a normal computational problem, and you could also meaningfully talk about pseudo-polynomial time if you, for example, consider numerators and denominators as separate integers.