How to append two bandlimited signals and make the result bandlimited without modifying the first signal
That's a tricky problem. First you will need to define how to design "integrity". What do you value more: time domain shape, spectrum, phase, envelope, short term Fourier spectrum envelope, etc. You will probably have to trade between those, so prioritizing would be helpful.

In general this is done bu doing "cross fades", i.e. you overlap the two signals for a small time period and then you fade out F(a) while at the same time fading in F(b). The time and shape of the fade can be controlled to satisfy the limited bandwidth requirement.

However, it seems you can't do this, since you can't modify F(a) at all. One potential work around would be to either extrapolate F(a) or use a  circular repetition of F(a) to create enough data for a cross fade. This depends on the exact properties of F(a)