If its possible to build a Gilbert-Cell Mixer using discrete components
Guess what Gilbert did! Yes, that's possible :)
and get it working on either a breadboard or a PCB
yes. Note that at frequencies this high, solderless breadboard is probably a bad choice, because the parasitic effects of the long metal bars inside will be larger than the values of some of the capacitors or inductors you need.
Note that discrete transistors are inherently bandwidth-limited. You will need to read the datasheets of your transistors very carefully to make sure your $\beta$ is high enough at the frequencies you need to work at! As a general guess: if it's available in a through-hole component package, its gain-bandwidth product will probably not be high enough for conversion in the 100 MHz region (and this kind of rules out soldered bread board). If it has "TIP" or "BC" in its name, it's going to be too slow. (Also: Get transistors for which you have a good simulation model, see below.)
Resources for design of Gilbert-Cell Mixer (I did try googling, but not sure which are the reliable sources.)
I'd start with Wikipedia (even if that muddies the waters a bit between what Jones designed and what Gilbert designed). It links to Gilbert's original paper, which is honestly easier to understand whilst still being correct than a lot of hobbyist literature out there.
Other than that, learn to use a circuit simulator! Gilbert did a lot of analysis with a pen on paper, and probably verified things numerically (with pen and paper and a lot of calculations); you don't have that time, but you have access to SPICE software, so learn that. You will not want to build circuits with transistors before being sure you're operating them all in the right point.
Also, plan for multiple prototype iterations: First you go for individual transistors to prove the concept, and be able to probe the signals using your oscilloscope. (You need to have some way to measure what you're doing!)
Second iteration would be more compact, and probably replace the individual transistors with matched pairs in the same package (for thermal equalization, see Gilbert's mixer paper referring to his own amplifier paper on having to make sure an equal amount of self-heating happens in both).
Do you have any suggestions for other mixers that I could use.
A couple (maybe 6 or so).
But for you, I think there's only one sensible other architecture: That would be a switching "chopper" mixer, which uses a square wave as multiplying waveform instead of depending on nicely linearized components for a continuous multiplication.
(there's many others, really, and I wonder whether you might have forgotten to read the "obvious" sources like wikipedia on frequency mixers, but you want an architecture that works well without too much tweaking. For example, a ring modulator with a high-power LO can be understood as very similar to a switching modulator, but you will need to understand how to measure, filter and isolate LOs if your carrier is sinusoidal, because diodes in real world aren't ideal, and you explicitly said you don't want to use an integrated component, where some manufacturer took care to choose transformers, diodes, and compensation elements for you).
I would call a switching mixer easier to build (at least approximately), and square waves are easy to generate well with CMOS technology (at least at the edge qualities that you would need), but you need to think about harmonics (after all, you might not even be using the fundamental of your square wave as LO!). But often, that's no problem, because your FM receiver is probably a superheterodyne architecture, so you choose an appropriate IF and filters, anyways.
However, fast edges then need you to put some amount of care into board layout, and might mean a few more transistors are necessary for high gains. I've never thought about building CMOS circuits from discrete transistors, as the "elementary" component no longer is the single transistor, but at least the complementary transistor pair, or rather the logic gate (and integrating that into an IC is a very good idea, for bandwidth reasons).
