Regarding building nuclear weapons. It affirms or strengthens the influence of Russia as modern power. It obviously does not support your nation to weaken your military because when you look at Greece, I believe had they spent the money to strengthen their military, no one would question that the Turkish invasion of Cyprus was no less illegal than the Greek invasion of Ancyra. Greece, is spending less and less and has cut American aid for a long time, yet for some reason Turkey is given billions of dollars to cover its continuous failure in economic policy by spending over 1/3rd of its gross national product on military.
Point being, Russia raises a strong military force for political influence. Turkey raises military force for political influence. Russia has maped the Transcaucas States under Russian territory, Turkey mapped (with the Soviets) Pontus/Armenia/Kurdistan and regained Ionia/Troas/Bythion/Thrace/Constantinople because of its military power.
What use does Russia have with nuclear power? Because obviously land forces are not persuasive enough in Groznyy, and maybe Russians want to stop dying by standing guard in a city the cannot protect by being present. Strengthening their influence can also persuade Chechens that they will profit more under Russia than by independence.
The problem with convincing these tiny states that independence is not good for them, is because before Soviets/Persians/Turks/Mongols attacked them, they were very rich countries, they won all their battles against Mongols/Romans/Greeks/Assyrians.
The Georgians are a mixture of ethnicities anyways? And even though they still have instability at times, there are still great prospects for the nation. Ossetia has been given great autonomy, as well as it is in Russia.