After playing through the whole game this way, I can report that the answer is 161 on Ultra Sun, or 162 on Ultra Moon (though the in-game dex only shows Alola dex entries, so it will give a number that's lower by 15). You can follow along with the logic every step of the way, including what notable things have to be skipped, at https://www.smogon.com/forums/threads/ultra-sun-dexit-challenge-seeing-as-few-pokemon-as-possible-complete.3669441/
If you cared about minimizing owned count, while there are no mandatory gift Pokemon other than the starter, in order to open the gate into Po Town you are required to partake in a double battle (if you only have one Pokemon, you simply don't get to make progress past that point until you genuinely do bring a second healthy Pokemon along). If you're keeping your game free of trades or other inter-game connectivity, this means you have to catch a second distinct species just for the purpose of taking on that battle, and can proceed to the end of the game with just that count of 2. However, if you do allow trades, you can pick up the Eevee egg from Paniola Ranch and trade it off before it hatches to get a second copy of the same starter you chose, allowing you to participate in that double battle while still keeping your Owned count down at 1. Alternatively, you could import your second copy of the starter from Pokemon Bank, and even go up to a full team of 6 copies of the same unevolved starter if you're doing it that way.
In chronological order (species marked with ^ are dependent on the particular starter you chose, but the count remains the same for any possible starter; I chose Litten so it'll be written from the perspective of that):
Up to First Grand Trial (21 US, 22 UM)
Litten^
Yungoos
Rowlet^
Spearow
Grubbin
Buneary
Pichu
Metapod
Bonsly
Grimer
Ekans
Popplio^
Zubat
Smeargle
Cutiefly
Drowzee
Gumshoos (Ultra Sun only)
Rattata (Ultra Moon only)
Raticate (Ultra Moon only)
Furfrou
Machop
Makuhita
Crabrawler
Up to Second Grand Trial (38)
Mime Jr. (Ultra Sun only)
Smoochum (Ultra Moon only)
Dartrix^
Zorua
Type: Null
Wishiwashi
Araquanid
Poipole
Marowak
Magmar
Lurantis
Salandit
Eevee
Raticate (Ultra Sun only, as Moon players already had this earlier)
Golbat
Anorith
Lileep
Lycanroc
Up to Third Grand Trial (59)
Houndour
Elekid
Electabuzz
Togedemaru
Golisopod
Masquerain
Bewear
Gastly
Haunter
Gengar
Mimikyu
Salazzle
Gible
Pawniard
Scraggy
Spinarak
Trubbish
Pinsir
Sableye
Krokorok
Persian
Aether Paradise (74)
Hypno
Muk
Magneton
Porygon2
Huntail
Gorebyss
Raichu
Houndoom
Manectric
Ledian
Claydol
Pupitar
Bruxish
Vikavolt
Clefable
Up to Fourth Grand Trial (103 US, 102 UM)
Fomantis
Mareanie
Beheeyem
Banette
Graveler
Lapras
Talonflame
Wailord
Glaceon
Noctowl
Flygon
Slowking
Jangmo-o
Hakamo-o
Kommo-o
Necrozma
Mawile
Granbull
Ribombee
Gumshoos (Ultra Moon only, as Sun players already had this earlier)
Komala
Trevenant (Ultra Sun only)
Tsareena (Ultra Sun only)
Shiinotic (Ultra Sun only)
Lanturn (Ultra Moon only)
Cloyster (Ultra Moon only)
Arcanine
Magnezone (Ultra Sun only)
Golem (Ultra Sun only)
Absol (Ultra Moon only)
Golurk
Mudsdale
Gastrodon
Up to Pokemon League (133)
Crobat
Zoroark
Lucario
Silvally
Scyther
Malamar
Vanilluxe
Mismagius
Froslass
Palossand
Dhelmise
Drifblim
Braviary
Hawlucha
Mandibuzz
Toucannon
Oricorio
Klefki
Dugtrio
Magnezone (Ultra Moon only, as Sun players already had this earlier)
Bisharp
Metagross
Armaldo
Cradily
Gigalith
Probopass
Crabominable
Vaporeon^
Decidueye^
Tauros
Noivern
Episode RR (161 US, 162 UM)
Blacephalon (Ultra Sun only)
Stakataka (Ultra Moon only)
Primeape
Clefairy
Scrafty
Shiinotic (Ultra Moon only, as Sun players already had this earlier)
Fearow
Tentacruel
Mightyena
Weezing
Camerupt
Groudon
Sharpedo
Kyogre
Arbok
Honchkrow
Weavile
Dialga (Ultra Sun only)
Palkia (Ultra Moon only)
Mienshao
Pyroar
Gyarados
Xerneas (Ultra Sun only)
Yveltal (Ultra Moon only)
Cofagrigus
Bouffalant
Hydreigon
Zekrom (Ultra Sun only)
Reshiram (Ultra Moon only)
Rhyperior
Nidoking
Nidoqueen
Mewtwo
