No, absolutely not. The significant part of the proceeds from that flow aren’t going into economy (specifics of selling oil to India).
The political situation is stable for following reasons:
1. Primary beneficiaries of military spending are small industrial towns and working class. They earn a lot of money now and significant part of it is invested in property or spent on domestic products. The inequality has reduced since the start of war, not something you would expect from oligarchic capitalism.
2. It became much easier to eliminate political opposition. Thousands have left the country, some were killed, many jailed under new wartime legislation.
3. There’s general perception that Russia is winning and it’s already in the endgame (which is true - the West lost the war in the first year).
The political situation in Russia is more stable now than before the war. Putin is certainly happy.