So I just happened to have really ridiculous Ballista only run (Had to choose some other tower towards the end, not useful choices at all) . This is just some next-level map RNG and also Mob RNG in that I got both Demons and Shapes. After few proper tries figured I would do better without March Mode for this Ballista only-thing, thought 1st Oogie is sorta close one and requires Frost Bolts 1 Card definitely... Anyways here's screenshot of that ridiculous map. I almost beat 1st Grim Oogie too expect regular mobs were bit too strong at that point.
But definitely pleased about this win actually! Other newer achievements should be way easier if I go for them.

https://steamcommunity.com/sharedfiles/filedetails/?id=2976171388

I ignored Universities, cause never had such card selection with them that I would want them. Took Seance very late too actually - that card didn't show up until very late. . Golden Forest was 2 barely worth just prior Phase 3 Oogie - seems I was either unlucky or % lied a lot. But it was almost clean win too, think like 1 or 2 mobs actually made it thru whole regular 45 levels :3 . From Phase 2 they didn't really stand a chance at all, due Map RNG.

https://steamcommunity.com/sharedfiles/filedetails/?id=2976171770

Congrats! What were your tower priorities?

Close to tower mostly Most Health - Fastest - Progress . As going towards outside I added mostly Most Armor - Fastest - Progress till round 11 added few (Antirams) Most Armor - Slowest - Progress. From 11 - 14 i bad RNG with Rams can end runs easily... You would want to max both extra health and armor cards ideally, with 1 Frost Bolts at the very least for Oogie. Add Ballista to +3 or +2 Hills that cover at least 2 Lanes if at all possible.

After that Left Portal I understood the Left Portal so could just add Most Shield - Fastest - Progress towards outside from that point on basically. Understood the other close Portal too by that time so it was rather safe to add mostly Most Shield - Fastest - Progress to that "common area" even thought it wasn't high hill close to that 3rd close portal. So the other close Portal mobs got deleted as they spawned basically. From phase 3 on any High Tower Ballista I upgraded 20 shield damage and any other Ballista 5 Shield damage. The Portal tiles have that High Tower pretty often actually so I uncovered those closest Portal tiles after 2nd Oogie (as only Phase 3 and 4 can have those high towers) .

Please note that this was mostly doable only due that layout and the mobs I had gotten thought. Might possibly add Slowest depending on how long road is and what mobs you have , but fastest tends to work best for me. I'm mostly meaning mobs that teleport after depleting shields. I would probably have gotten away with Elementals too tbh, but Robots and particularly Invaders aren't really doable without any kind of AoE. For Robots you'd need really forward Most Armor - Slowest hitters for their missile spawners asap, for Invaders their tanks are super harsh so idk if thats like doable at all with just Ballistas.

Thanks for the detailed response! I have been trying to knock out these achievements and yeah, it's difficult lol. Crazy that you did it on four lane