Yes, the unusual word order in "should that interest the president" is an example of conditional inversion.
But if one tries to make it more comprehensible by simply reverting the word order to the more familiar "that should interest the president," the meaning of the phrase is changed. In this new arrangement, the speaker is now declaring that service ought to interest the president. In the original phrase, the speaker is uncertain about the president's interest.
To change the word order yet keep the same meaning, one needs a way of expressing this irrealis. That's the subjunctive, which is used for hypothetical situations, things that possibly may or may not exist (like the president's interest).
Another way to phrase it (in a familiar word order but preserving semantics) could be to say, "It would also be doing a big service to the honest business community and country, if that interest the president as well."
However, in everyday spoken English the subjunctive form for many verbs is increasingly uncommon (except for be and were ). So it may be aceptable in some circles or even preferable to use "It would also be doing a big service to the honest business community and country, if that interests the president as well."