First off welcome to the forums.
2nd, we don't really use the word best here, in fact it is a forum rule/guideline take a minute and peruse it, you will find a more intelligent discussion if you can phrase the question more precisely. One could make a numerical argument for something over the other, but best is a matter of opinion and is highly subject on purpose, use, budget, place in life, location, etc etc etc. There really is no best, semantically there can not be a best of two options, one can only be better then the other, once again still highly subject.
They are very different engines that deliver power differently, it will be up to you to decide what you want. An LS3 could probably be pushed to 400 or so HP NA I imagine. Linear torque curve and all. The 2jzgte can hit 400 on the stock twins which are a sequential setup, a 2jzge NA-T could hit 400hp with a small single turbo. An LS3 swap may be easier to start, as the 2jz option only has one really good option for an off the shelf kit, which will set you back quite a bit more compared to an LS swap kit. The 2jz can be easier to add hp depending on your turbo selection and injector size it would be a matter of turning up the boost. Where as on the LS you might have to start considering forced induction if the power bug hit you.
Does it have to be manual?
The 2jz manual options can be pricey. The V160/161 occupying over half of your budget with a good clutch. The CD009 while being much cheaper is still quite a bit in the conversion/adapting hardware. The R154 while also substantially cheaper with upgrades and a twin disc is still a good amount of your budget. The W58 can be a very cheap option, but also is known for not holding too much torque and power, not really a long term option.
Easiest I imagine would be going from chevy to chevy. Someone else might be able to chime in regarding the mounts and stuff between a 350 and an LS. Any reason you don't want to push the 350 to 400hp? That seems like a lot of money to save a bit of weight with the LS if that was the route to go. I imagine new heads, different intake, maybe even a supercharger or blower would put you in that range with very minimal work.