Speaking from experience as someone who bulldozed through the story and got done in time for the event:
generally speaking, if you can get through most of a boss fight that only a marginal sliver of health is left on said boss and you can spare them, while your Command Spells are on CD spend a crystal. Saves time unlike when you go out of your way and try and grind just to get past said boss.
On that particular fight, I myself did do that exact same tactic you suggested: borrow a bond 10 Herc, take down Ozy’s Caster side-kicks and then hope that my mediocre-ish Assassin-game can carry the day once Herc keels over. My fully ascended Long Arm Hassan managed to take Ozy down to a couple thousand HP in the end. Having taunt-options helps divert damage from your Herc and 'sins, I spot two of those in your roster (four including Lips and EMIYA Assassin, need ascension though). If you happen to have a CE that applies taunt to a Servant once they hit the field, you can use that too on the back-line to some great effect - do level David, whom you’ve received in Okeanos, to level 30 and ascend him once, he can serve his part even then by both applying his party-wide dodge and eating a couple guaranteed hits, buying your main team some time.
Camelot is rough, particularly when you do not have a fully upgraded team. Difficulty-wise, Ozy’s battles both in tentacle- and normal form has been my personal zenith there, so once you get past that you will face the Knights of the Round Table going through their familiar motions once they show up again. Their boss, Lancer Artoria, can be nasty, though, particularly in the 2nd fight what with her Invincibility-Pierce on AoE-NP and her even greater stupendous amount of HP. Try and redirect her normal attacks to taunters, have a couple strong Sabers dish out damage from the get-go, maybe even keep one in the back-line and you should be able to plow through her HP. Alternately, go full zerk in the first row and have those do some damage, with Sabers coming in later to hopefully wipe her out. Again, if your team gives out with most of her HP bar down, consider spending that crystal.
If it’s any consolation, Babylonia will be less tricky at first, with just general stats on enemies being beefy. 2nd part of it shakes things up quite a bit, however (also, the feels…poor Urukeans). And, looking further ahead, for the final couple bosses in Solomon, consider getting some form of Evasion pierce or buff removal - raise Medea to max, her NP is gold for clearing Evades for other NPs to hit and readily available with her kit.
Hope you’ll enjoy things irregardless of difficulty spike. And there’s always the re-run of the current event somewhere down the line. :S