Applying multiple IIR and FIR filters at once
When someone says “I want to do x” , it helps to know why? 
Instead of going into why you want to do your “x”, it might be instructive to go into why it is probably not a good idea.


There are some filters where there are  mathematically equivalent recursive and non recursive forms, but they don’t have the same robustness to rounding errors.
Combining recursive and non recursive forms into a single filter is possible. The numerator in a ratio of polynomials that typically make up a transfer function can be thought of as a non recursive component , so one can end up with a combined filter that is recursive.  You can also truncate the impulse response to end up with an overall non recursive filter. In either case it really can’t be both.
Polynomials are delicate. Small perturbations in coefficients can make big changes in roots, particularly when those roots are poles near the unit circle. Analytically, it can be stable but in practice it isn’t. In most cases, we factor transfer function is structures like biquads. Lots of cascades little filters are almost always better behaved.
Humans delight in their cleverness, but very few things are universally acknowledged as being cleaver. In practice, if this filter is “product” someone else will come along and maintain your code. In most cases, that person ( which might be you after a few years) will not share your joy. They are more likely to curse you and if you are very cleaver, they will curse your mother and father. One should chose clarity over cleverness.
You really need to do something a number of different ways in this sort of framework to make tradeoffs. The term “exhaustive enumeration” applies to this
problem. The different ways a band pass can be implanted is really not hard to compare most of the
salient parameters. You have to appreciate how much harder you make
your filter optimization
When a parent gives their child a bicycle, it will take a number of years before they enter X-Games. Given that you asked the question, the ultimate answer, is yes but will require great skill.


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“Whatever doesn’t kill you makes you stronger” I encourage you to follow up on your ideas. Try to make it work. Endeavor to make me wrong,