Damage and fire rate tested against Hazuth docker (Gunner fish)

Tier 1
Hysteria
DMG: 8-12, fires from above and can be blocked by roofs.
Fire rate: once every ~3-4 sec
Location: Hazuth upper you whould see it on your way to the first boss.

Brain in a jar
DMG: 8-12, fires from the front
Fire rate: once every ~3-4 sec
Location: The Engine, right side hidden wall, below main hub

Lens Reaper
DMG: 8-12, fires from the front
Fire rate: once every ~3-4 sec
Location: The Engine, Given on your way back to hub from queen (boss 2 defeated)

Tier 2 (After false queen)
Arctic fox
DMG: 13-19, spawns on target but only on ground, can't be used for flying bosses.
Fire rate: once every ~3-4 sec,
Location: Below london, need teleport. (boss 4 defeated)

Umbras fox
DMG: 11-15, fires from the front
Fire rate: once every ~3-4 sec
Location: Top of Cathedral Belltower, need Teleport. (boss 4 defeated)

Kings One Wish
DMG: 13-20, calls a lightning bolt down on enemies, can stand on a level below them and still hit, good reach. but often miss moving enemies. Can't harm shoggoth, dosen't count as lightning element.
Fire rate: once every ~2-3 sec.
Location: Behind mirror in library. (Boss 5 defeated)

Parasyte Waltham
DMG: 8-12, fire from the front.
Fire rate: once every ~4 sec
Location: Obtained after defeating Shoggoth, need teleport.

Any of the front shooters should be good to use for lvl 1 only runs, as Umbra fox has slightly more damage than the others it could be a good choice though you get it a bit too late so a better choice would be Brain or Reaper. Kings and Arctic aren't reliable enough for bosses as they both have problems with actually hitting them.
Walthams stats speaks for themself considering you can only get him near the final boss (Electric whip) and he is more or less the same as tier 1 pets.