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I've been told that synthetic ultra-heavy elements are produced by bombarding a light nucleus into a heavy one, and that several combinations are (at least in theory) possible, as long as the sum of their atomic numbers matches the one of the desired product.

Then, how do they choose among these different possibilities?

For instance, this video states that Livermorium atoms were produced in Russia using Calcium and Curium, but the Curium had to be imported from the US. Why not use a different combination then?

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To choose among different possibilities, theoretical calculations are made to evaluate the likelihood of a successful fusion between the candidates.

This is not a simple evaluation because it relies on nuclear models, and several branches may occur. The FIAS institute in Germany collaborates in this subject with Dubna in Russia, and they have made estimations of the likelihood of creating the desired nucleus. They used a model called the Two-Center Shell Model to study the phase of the compound nucleus (which is the initial stage of the system formed by the merge of the two nuclei). With those studies they were able to suggest "paths for cold fusion", which are paths in phase space which accounts for shape, momenta and asymmetry of the starting nuclei.

With this information you can say what pairs of nuclei are better because you can estimate the cross section of the compound nucleus formation. But furthermore, some studies are needed because the compound nucleus may decay in different ways, and estimating the relative probabilities of these decays helps in setting up the experiment and detecting if the nucleus actually was formed.

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