Well it depends on what peripherals you have installed or planning on installing to your car later.
To properly size a turbo you need to have some idea on what you plan on doing.
KKK Turbochargers:
KO3(S) - 200-220 whp 230-250 wtq
KO4-001 - 220-240whp 240-
280 wtq
KO4-20/22/23 - 230-260whp 200-220 AWHP 260-300 wtq
All these turbos are good for TRQ producing. They are great for lower band power output (2500-6000 rpm) but most of the times after 5500rpm they have major drop in power. They have reached their limits.
Garrett GT-R / GTX Series Turbochargers:
GT turbos are dual ball bearing if they end with a R. If not, they are journal bearings and have gt style wheels. DBB turbosspool up 15% faster and consume less than 1/2 the power than journal bearings turbos. This means faster spools ups and more over all power than journal bearing turbos. Currently the GTX turbocharger series is available and produces 10%+ higher pressure ratio compared to traditional GT compressor wheel designs, 11 Full-blade design for improved efficiency and ultra quiet operation (as they advertise it

), Forged, fully machined wheels (billet), interchangeable with Garrett GT Series turbos. I don't recommend GTX due to very high costs.
For 300bhp i would recommend you:
GT28R /GT2560R - 270-300whp - 3000rpms
GT28RS /GT2860R -
290-330whp - 3200rpms
All housings are based on .64 T25 housings for the GT28 family and .63 T3 housing for GT30/GT35 turbos. These work out for the fastest spool with good flow.
T Series Turbochargers:
These turbos are all journal/thrust bearing turbos unless the chra is changed to a dual ball bearing unit. They are often just oil cooled unless upgraded. You can also get the T series turbo in two versions 270 or 360 thrust bearings. The 360's oil far better and since the T series turbos use more oil, it is a good idea to get the upgrade. If it doesnt say 360 it is a 270 thrust bearing. Journal bearings spool slower which isn't always a negative. A T3T4 turbo can make more power on the stock rods than a faster spooling GT turbo. Not to mention that since they use older tech, they are about 1/2 the cost of a GT turbo. You can however decrease lag on a T series with a smaller housing (using a .48 instead of a .63).
For 300bhp:
T3S60 - 270-310whp - 3100rpms
ATP Eliminator Turbochargers:
These are modified GT turbos with a K03/04 style housings. They are a great concept but the
problem with these turbos is that the flange is a huge restriction on flow. Also the housing style doesn't flow well enough for a high hp turbo. The GT2X and GT28R will be fine using a upgraded manifold and have reasonable spool. The GTRS and GT71R have a rather large increase in lag and lower power output compared to the T25 manifold and housing. Also it sounds like it's cost saving but in the end, it costs close to the same as a whole kit.
GT2X- 240-250 whp stock like spool with more psi held till redline
GT28R - 260-
280 whp - 3300rpms
GTRS -
280-300 whp - 3600rpms
GT71R -
290-320 whp - 3900rpms
But for budget side choose wisely and tend more to T series turbos due to lower costs or KKK turbos. But you will also need some other parts for this recipe. I am not talking for Pistons and Rods as you already stated that you rebuild your engine. I would recommend you cast iron manifold, FMIC, 4 bar FPR depends on the tuning software that you plan installing Unitronic, Revo, APR,GIAC, Injectors at least 630cc Siemens Deka if you plan on making bigger power numbers on the future than 300bhp or 550cc Bosch injectors for the power that you plan on making right now which is arround 350bhp but i believe that for 350bhp the 550cc will be maxed out. Also you will need high impedance wide angle spray pattern injectors. If not, your idle and fuel mixture will be bad. Please stick with Genesis and Siemens Deka injectors.