Why wouldn’t Putin do that? It is such an obvious gambit.
It is a bit risky because such activity could cause a crackdown by the Moldovan government. However, I think he is correctly reading the government as unwilling to crack down. So if he can topple the government and establish a Russian-friendly government in Chisenau it may cause the Ukrainians to divert forces from the eastern front. It is risky because if the Ukes feel threatened enough they could potentially attack the Russian-friendly government before it can even set up its first concentration camps. And they would roll over Putin’s Transnistria ally in a heartbeat.
If it works, he has a new threat on the rear border of Ukraine. If it does not, he loses Transnistria. But he may have decided that the potential sacrifice of an ally is worth the gamble.
It appears that the gambit may not have worked. But the Moldovan security forces are probably very busy right now.