Bit of a novel upcoming response but hope the reason makes sense. Also bear in mind while so of this is fact, alot of the 'better' options are personal preference and there is no right or wrong answer.
whp is typicly US value and is major over eged.
Fly wheel BHP is the typical measure, at the road you loose roughly 23-25% (very rough figure, alot of things affect it condition of gearbox, oil thickness in the gearbox, any loses through the diffs etc etc) so 550bhp at the wheels will be 700bhp at the fly based on a 25% as an easy figure.
What the stock engine will do is very dependant on where the starting point is. A jdm is 2.0 and has proper forged pistons so will prob do 450 with bolt on's. 6 speed box has the synchro weakness as you say, but typicly not an issue.
But there's also more to it than that. Car use has a big difference to how you would achieve it.
A race car will be belt and brace, possibly Linered closed decked etc etc masive list of supporting mods, coolers, etc etc mega overspeced and then run at lower power to try to retain reliability. Most top race cars, eg gobstopper, have a refresh after every event and can still fail. They typically run at high revs and high boost so generate alot of heat and wear quickly.
A road car is a very different kettle of fish. You need to look at matching gearing and engine. If they mismatch you could find the ratios mean the turbo can;t spool up before your changing gear (i had this issue on mine when i was running a 2,5 on short ratio 5 speed box, the torque the engine was producing meant the turbo couldn't keep up so the car pulled harder in 3rd and 4th than it did in second)
For a road car there is a phrase, 'there's no replacement for displacement'.
So to answer your question,
450bhp at the fly is a good figure for the road. A 2.5 is my personal preference (why i went that route), they produce considerably more torque than a 2.0 so have power lower down the revs. EG 2.5 400bhp/450fltb, 2.0 400bhp/350ftlb or a mix there of. I'm running 398bhp/465ftlb. Combined with the right ratios (im on a uk hawk DCCD box) they are fantasic on the road.
So let say you go for 450bhp on a 2,5. A reliable 450 which is the key to it
Your gonna be looking £4.5-5k for a built 2.5, forged piston, rods, head gaskets, acl race bearings, arp headstuds. (possible additional if heads need rebuilding).
Then you need to look at what else you need to get there, it will only perform as well as the weakest part,
Option to closed deck the block £500
Turbo £2k
Front mount £2k
Injectors £400
Fuel pump £200
Inlet pipe £250
Decat £600
syvecs (ECU struggles to cope at this level) and other options are available, but IMO not as good, £2500 inc installation and mapping
Clutch £1000
fuel pressure reg £250
3 port boost solenoide £100
4bar map sensor £200
£13,000
then add in all sort of ancillarys that are advisable to do, oil and filter, water pump, oil pump, cambelt and tensioners etc etc
and your prob up to £14k give or take. works out alot of money when you look at whats involved.
Pick up your chin now lol
Can you get there cheaper? yes you can, will it last as long and be as reliable? 9 times out of 10 nope.