How can I make a very temporary "backup battery"?
The real problem seems to be that your battery can't run everything you are trying to make it run.  The best fix is to address that directly.  Get a bigger battery or less stuff to run.

Does it really need to be a 96 W car horn!?  How much noise are you really trying to make?  Chances are that you don't need that much noise, and/or that there are more electrically efficient ways of making such noise.  For example, there are some self-oscillating "buzzers" that are still quite loud but take a lot less power than that.  Then there are also things like fire and burglar alarm horns that are meant to be run from backup battery, so probably don't need 96 W to make a lot of noise.  96 W is no big deal in a car, so efficiency was likely not much of a design criterion for the car horn.

In any case, it is probably a good idea to put a Schottky diode in series with the power feed to the computer, followed by a cap to ground.  How big the cap needs to be depends on how much current the computer draws, which you haven't told us.

Another strategy is to charge up a cap slowly to be used only by the horn.  When the energy in the cap is spent, the horn goes silent.  That will likely give you a shorter horn burst (with this pig of a horn), but it won't be able to load the battery to the point it can't power the computer anymore.  It will also take some dead time between horn blasts for the cap to recharge before the next horn blast is possible.  This is very much like the concept of a camera flash having to charge up before use, then having to wait for it to recharge before it can be used again.