Day Start and Must Lynch means the Gallis is clear and should out, If the Sniper shoots before the Gallis claims they're not thinking whatsoever.
If the Sniper shoots the Politician they have lost the game and may have dragged the Politician down with him.
A Town aligned Politician should always try for mislynch as it always improves their chances to win, Inversely a Mafia aligned Politician is very vulnerable to a town mislynch if it isn't the final lynch.
As long as the Gallis is alive the Politician and Sniper can nolynch to force their alignment to ether side, This should be done for the side that offers the Politician more points which is Mafia.
If anyone claims Politician Day 1 before the Sniper shoots then everyone should lynch them instead of the Gallis, This is nether good or bad for the Town or Sniper.
If the Sniper shoots Gallis everyone massclaims and a lynch between Snoop counterclaims occurs, If the real Snoop is lynched then who wins will be chosen by Politician alignment which gives Town a 2nd chance.
If Politician claims when Gallis is shot then everyone including Sniper should lynch them, As the Town will lock votes with Politician and Sniper thus nolynching forever. Making the Politician useless to the sniper.
If the Snoop is shot then the Politician should claim Judge alongside the Sniper to try and get the Town to mislynch. Trying to lock votes to figure out who is Politician or Sniper works but can backfire with Gallis Pop. The Sniper is doomed if the Politician is lynched after Snoop is shot.
If the Judge is shot then the Sniper and Politician regardless of alignment out themselves and lynch the Snoop who should be counterclaiming Sniper now.
If Sniper doesn't shoot Day 1 then Day 2 the Sniper must fake a Snoop report, If ether report nether clears or confirms the other of having a gun then Town lynches between the 2 Snoop claims. The Judge and Politician need to counterclaim each other to avoid being shot.
When the Snoop and Judge dead from ether gunshot or mislynch in any order and the game is down to Sniper, Politician, and Gallis. The Sniper and Politician can lynch the Gallis until the Politician is Mafia aligned allowing both to win points when the Politician selfvotes or pops the Gallis.
The Politician doesn't get a win notification without points, So without points the Politician loses all incentive to fight for any side completely throwing the balance of the setup. Politicians who don't side with whoever offers more points when they have the choice are gamethrowing this setup.