The aircraft you're describing has absolutely nothing to do with Hürjet, you're talking about a completely new aircraft. At that point, we'd be far better off with starting with Kaan's design instead of Hürjet and making a smaller, single engine Kaan for the navy.The ideal solution would be a low-cost, naval stealth aircraft. The Hürjet M should be 20–30% larger, with significant design revisions incorporating stealth features, drawing on elements from the TAI KAAN. With two F-404 engines, it would have sufficient thrust to carry more payload while remaining agile and fast. A small internal weapons bay capable of carrying 2–3 air-to-air missiles (AAMs) would allow it to serve alongside Kızılelma as a stealthy interceptor. In essence, this revised Hürjet would resemble an enhanced F/A-18 Hornet, but with better stealth characteristics and modern systems.
And as it is, with minimal changes, Hürjet is a shitty choice for an aircraft carrier that we expect to have in 10-15 years. It is already behind the times, by the time it is on the carrier, it'll be completely obsolete and nothing but an embarrassment and potential missile fodder.
It's a bloody trainer, its inherent weakness come from the fact that it is a trainer. As long as you're basing something on it, that version (whether it is light attack or naval) will share those weaknesses.