The first issue is you must use a conductor type that is listed in 300.13 if you want to install it in a raceway or a flexible cable type that is listed in 400.4. However flexible cables listed in 400.4 are very limited when it comes to installing them in raceways.
Next comes ampacity.
The manufacturer can rate it any ampacity they see fit, we have to ignore that rating and use the NECs ratings which will likely result in the 75 C column of table 310.16 or 150 amps for 1/0.
While we are at it, yes, table 310.16 shows 1/0 THHN as 170 amp rated in the 90 C column however none of your equipment terminations will be rated higher than 75 C so 1/0 THHN will be effectively good for only 150 amps.
1/0 THWN/THHN 90C allowable ampacity is 150 amps and will need over current protection at 150 amps.
1/0 flex-a-prene -50C to +105C heavy duty welding cable if it is listed in 310.13 also has an allowable ampacity of 150 amps and will need over current protection at 150 amps.
There is no good reason to use welding cable for this, it will pull hard in the raceway and the termination on your equipment are not designed for the fine stranding which means you will have to use a compression tool and connector.
I install 225 amp panels quite often and at a minimum we use 4/0 copper with a 75 C rating of 230 amps this allows the use of 225 amp overcurrent protection
3/0 copper is rated 190 and the largest allowable over current protection will be 200 amps.
Bottom line is 1/0 of any type does not come close to providing 225 amps.
Most systems work best if they are kept simple rather than made complicated; therefore simplicity should be a key goal in design and unnecessary complexity should be avoided.