I’m slightly interested in swapping in a 5.3 into a ‘71 Cutlass.

I’d prefer to run headers and retain the BOP TH350. It’ll be carbureted as well. I’m aware I’ll need an ignition box to fire the coils.

Where/what brand should I source parts for the swap?

 

Keep the Olds, having been there, doing that, unless you are planning to make stupid power just keep the Olds motor and if you want it running smoother buy a Holley efi sniper kit and replace the carb, call it a day.

Leaving a TH350 behind a LS is stupid, the sole "reason" for going LS if not outright horsepower is for a smoother more modern feel. An antique 3 speed limits it greatly. Doing a 4l60e or even a newer 10speed would make infinitely more since.
SInce you want to do it cheap with a 5.3 and leaving the BOP Th350 you probably don't have the money to really do it.

If you are absolutely hell bent still for whatever reason here is the complete list of stuff to do this nearly as cheap as possible.

• Buy a 5.3, pre-dod and with harness and ecu
• LT1tech has been around since people were swapping the OG c4 LT1 engine. He will reflash your ecu, remove vats, and get you running cheap.
• LT1Tech has the pinouts to DIY your own harness that came with the engine.
• Buy a new fuel tank for EFI, you old one has 60years of garbage just buy a new one. Tanks Inc is affordable and has the full kit.
• Run new lines. Braided garbage is a pain and overpriced for your use, also dirty secret they don't tell you is those braided lines are unlined rubber and ethanol will destroy them unless you spend even more for the Teflon whatever ones. Use Russell synthetic rubner fuel lines, no braiding. You will need adapters to the factory rail.
• Buy a LS Swap Radiator, strong suggestion. Again your old one is old buy a new one, avoid a 5 foot hose making a u turn because a truck water pump output doesn't face the right way.
• Buy a splice for the radiator hose for the steam line.
• Pick some mounts, doesn't really matter.
• You need a VSS pulse generator from Dakota digital for the engine to know it's moving/underload.
• Speed engineering headers are cheap and fit perfect.
• You need a Power steering Pulley off some 90s GM junker, maybe of a 3800 series? For the truck power steering to clear.
• Electric fans
• Size out a radiator hose
• Size out a serpentine belt
• Splice an ignition hot to the ecu harness

turn the key and now congrats you have a fuel injected 400hp V8, that you could've had with a cam swap and Holley EFI on your olds engine at 1/8th price.

edit, just noticed you wanted to keep it carbed. Now really gotta ask why? What's your power goal? Everything above still applies accept the fuel tank.

You will need a Chevy to BOP trans adapter and either a custom converter or a snout extension/adapter also for a th350 behind an LS.

 

Avoid the new GM 10 speed transmission. I'm not even sure if an aftermarket company makes a controller for it. It's a joint transmission with Ford and has had horrible reviews.

 

Maybe I was thinking about the 8 speed, I know TCI has aftermarket support on it for sure. Surprised to hear anything negative on the 10speed. I've driven a few vehicles with it and loved it. Either way, just stay Olds unless you are trying to get north of what the blocks can handle without band aids.

 

The Ford software for the 10R80 is worse than the GM software. I agree with everything else that you mentioned though. The easiest upgrade would be a Holley Sniper EFI kit.

 

If I read OP correctly, this car will have an LS swapped into it, which will subsequently be converted to carburetion.

WTF?

 

Yep, horrible idea. A carbed LS with a TH350 and probably 2 something gearing, barf. GM's 8 speed has plenty of issues as well. The 4L60E is a steaming pile under any power. An EFI 6L/4L80E is the only worthwhile swap. Price everything out, won't be cheap. There is limited support in the aftermarket for the later electronic transmissions. Plus good luck in fitting the huge 8 and 10 speeds under your 72.