I was thinking: we have a small cache of alloys, allowing a single national project using them.
Our main problem here is that we lack the technical and/or scientific knowledge to do most things that would be useful against aliens, but alien alloys are the holy grail of material engineering: great resistance to everything alloys, room temperature superconductors, etc.
This means that if we pick a design with a very simple design but which wasn't used at the time ( or even now) because materials were not suitable, we could achieve a great leap in our tech.
My idea is railguns. The underlying physics is trivial and well established. the design is very simple. The problems are rail wear and energy storage/release. We can solve the first using alien alloy rails and the second with superconducting magnetic energy storage ( basically, a coil of superconductors with tons of current going through it, storing energy in the magnietic field), which is made significantly easier by having room temperature superconductors available.
Power generation would be supplied by the quite inelegant solution of mounting it near a power plant. or a suitably large generation.
If we mount the railgun on a turret ( power generation stays on the ground clearly), we could have a decent chance of shooting down some UFOs, before they destroy or avoid it.
considering that the US navy proposed railgun was going to use 64 MJ and that SMES is extremely efficient, a well sized power plant should allow several projectiles per minute ( on the power side alone. I suspect that the power transmission from plant to SMES is going to limit us). A railgun might have a range of a couple hundred kilometers ( after at least one full turn, or maybe one design and 2 revisions) and accuracy problems may be "solved" by a small timed explosive charge on the projctile turning it into a cloud of fast hot pieces of metal in flight.
So, I believe this could be a good chance to start fighting the air war. advanced weapon, easy engineering suitable to the time thanks to alien alloy use, great performance. Can't think of reasons why we would not be able to build one with what I have at hand, as long as we can shape the alloys somewhat. We could build only one, sure, but with the range of a good railgun we can protect a sizeable area with it.
With that in mind, I switch my vote to Alien alloys II