Weenie 400
Kanto's actually not so bad for 400-BSTs, because Kadabra comes RIGHT at 400 and there are multiple strong Pokemon below that, such as Weepinbell and Graveler. Early on, you get Butterfree, which can carry you with Confusion until you get the rest of your team. (The only real issue is Porygon, who costs 130,000P... ouch. Unfortunately, it's the best Thunderbolter available.)
The major obstacles to victory are Lorelei's Lapras and Lance's (and Rival's) Gyarados, which is why a T-Bolter helps a great deal. Choosing Bulbasaur helps nullify Rival's otherwise-fearsome starter, as Charizard can get locked into Gen-1!Rage and has a 4x Rock weakness for Graveler to exploit.
Weenie 350
Staryu is basically the best Pokemon of the category, with a diverse movepool and 340 BST. Gastly is a close contender with its own movepool, its ridiculous Special and Speed, and useful immunities -- it should probably be the T-Bolter. Aside from that, you're in for a lot more pain, because even Butterfree is over 340.
Drowzee is probably your best Psychic because its goal is to block Alakazam. The importance of blocking Alakazam cannot be understated in Gen 1.
Diglett may be your best option for Ground, painfully flimsy though it is, due to its high Speed and early availability.
You'll again want Bulbasaur to weaken the enemy starter. Early, your team will probably be carried by Bulbasaur with Clefairy/Geodude backup. Clefairy is pretty tough in the early game, so that will probably carry you to Drowzee and Diglett.
Weenie 300
The absolute biggest issue, of course, is the absence of a starter!
The cheapest way of getting past Brock, based on a run of a No-EXP hack, is to get two Pidgeys to Sand-Attack level and burn one apiece on each of his Pokemon. Then Growl down with available Pokemon (in this case, a Pikachu), switch before KO, and Tail Whip and bash down with Rattatas. RNG-dependent, but better than having to grind for ages.
Past him, it's not so bad. Spearow can be employed for Peck, and Geodude squeezes in at exactly 300. From there, you use Pikachu for Misty and get Diglett for Surge.
You do hit a wall past, say, Celadon, because it's increasingly hard to pass the Gyms with 300 BSTs. Explicitly, you can only use Pikachu, Sandshrew, Poliwag, Bellsprout, Geodude, Dratini (56000P|92000P), Vulpix, Horsea, Meowth, Ekans, Ditto, Paras, Nidorans, Jigglypuff, Diglett, Spearow, Rattata, Pidgey, Zubat, Metapod, Kakuna, Magikarp, Caterpie, and Weedle.
I will personally go with Red, despite the much smaller pool, because spending nearly 40K less on Dratini means a lot more leeway cash-wise.
Final team is probably something like
Poliwag: Surf | Psychic | Amnesia | Hypnosis
Dratini: Thunderbolt | Thunder Wave | Wrap | Surf?
Pikachu: Thunder | Thunder Wave | Body Slam | Swift
Diglett: Dig | Rock Slide | Sand-Attack |Slash
Bellsprout/Ekans: Toxic | whatever
Dratini has that weird moveset as an attempted Lorelei/Gyarados-killer, with Pikachu for backup. The Poison-type is solely to shipwreck the AI for Lance's Dragons. There is no coherent strategy for Alakazam, save attempting to land Thunder Wave, Hypnosis, or Sand-Attack and going from there.
If I get too bored with the Elite Four grind, I can always break out X Items. X Speed x 30 and [Gen 1] X Accuracy x 5 could probably sweep the E4 for the measly sum of 15250P, because Pokemon can't attack on the same turn they wake up in Generation 1 and a fully boosted Poliwag would have a 100%-accurate Hypnosis with 270 effective Base Speed. Obviously I want to avoid this unless I'm verging on giving up the challenge otherwise, but it's a safety valve if I end up completely stuck.