Warpriests are the only full casters that do not get legendary proficiency in their spellcasting tradition.
Summoning does not rely on spell DCs in order to work.
Thus, Warpriests shouldn't be any worse than any other full casters at summoning. Normally, Clerics would not have access to low-level summoning spells outside of some very specific circumstances (Tupaliq Carver for Summon Construct), but with the release of Gods and Magic, there are quite a few options that provide access to different summoning spells.
Gendowyn (N, CG, CN, CE) - Summon Fey
Any of the Evil Elemental Lords (N, LE, NE, CE) - Summon Elemental (Air / Earth / Fire / Water only)
Chamidu (NG, N, NE) / Baphomet (CE, NE) - Summon Animal
Halcamora (LG, NG, N) / Xamen-Dor (NE, CE) - Summon Plant or Fungus
Shizuru (LG, NG, LN) - Summon Dragon
Nyarthotep (CE, NE) - Summon Entity
Any thoughts on specific synergistic combos that could work well?
Getting Wizard dedication for Augment Summoning or Bard for Inspire Courage helps the summon to be stronger if you want to save spellslots.
For summons I believe that Summon Fey might be the best one, have a good amount of fey with good spells like the Unicorn.
A large chunk of "buff" spells have been removed with second edition. However, clerics still have a fairly decent selection left at low levels (spell level 1-3):
- Bless
- Heroism
- Magic Weapon (Only Summon Fey would have applicable targets)
- Shield Other
- Spirit Link
There are a few ones after level 3, but you're usually better off heightening Heroism in most cases.
Given that, I'm not sure it's really worth it to multiclass into something else. The only exception might be a primal caster to get access to Magic Fang -- it would work with the majority of your summoned creatures -- but that would require two feats.
Summon Celestial, which is already on the divine list, is already one of the best summoning spells in the game (in no small part because of the Shield Archon). The biggest restriction is that they have to match an alignment allowed by your deity, which means that most evil clerics can't make good use of this yet.