Is it numerically more stable to implement filtering as multiplication or convolution?
With convolution, you are not going to run into any stability problems, because there is no recursive filtering, so you are not going to accumulate any errors. In other words, the system is all zeros, no poles. I have heard anecdotally, but not verified for myself, that FFT-based convolution has lower error than time-domain convolution, simply because it has O(n log n) arithmetic operations rather than O(n^2).

Typically, as far as I am aware, Butterworth filters are implemented as recursive (IIR) filters, so that is a different topic. IIR filters have poles as well as zeros, so there can be stability issues in practice. Also, for IIR filters, FFT-based methods are not an option, but on the upside, IIR filters tend to be very low order.

As far as stability issues with IIR filters, they tend to have problems at higher orders -- I'll just throw out a number and say roughly 6th order is pushing it. Instead, they are typically implemented as cascaded biquads (2nd order filter sections). For your 5th order filter, write it as a z-domain transfer function (it'll be a 5th degree rational function), and then factor it into its 5 poles and 5 zeros. Collect the complex conjugates, and you'll have two biquads and one first-order filter. In general, stability problems tend to crop up as the poles get closer to the unit circle.

There can also be issues with noise and limit cycles in IIR filters, so there are different filter topologies (i.e. direct form I, direct form II) that have different numeric properties, but I wouldn't overthink this point -- just use double-precision and it'll almost certainly be good enough.