Flight III Burke will displace more than a Flight II Burke, looking over 10,000 tons, and at least 45% of the ship will be new. Its a much bigger jump than Flight I to Flight II Burke.
Flight III Burke doesn't use AEGIS, because AEGIS is technically a Lockheed Martin brand name for their radars, and AMDR happens to be from Raytheon and therefore a competitor. Technically its also a different radar from AEGIS being an active array or AESA (similar to the Type 346 used on Type 052C/D and 055) while SPY-1D or AEGIS uses a passive phase array or PESA. The difference being is that every element in an AESA has its own transmitter and receiver, while in a PESA, every element has its own receiver but shares a common transmitter. So to speak, AESA is what OLED is to PESA which is LCD. What AMDR has is that its commonly compatible with AEGIS common libraries (apps, software).
The cost of Flight III is going to be high, probably about $2 billion to $3 billion. May end up more.
To make a new Tico successor cruiser that would differentiate itself from the Flight III, you will likely need at least a 17,000 ton ship, something closer to the Russian Lider class. This will be tremendously expensive and complex.
Given that the USN is now required by law (!!) to have a 355 ship navy, it no longer has the option of going with fewer but more expensive ships, but rather, instead with greater numbers of lower cost ships.
And that is why so much depends on the FFG(X) program to provide the USN with a proper workhorse that needs to fit between the LCS and the Burke. I prefer the expediency of smaller ships --- you can build them with not just less cost, but with shorter construction and fitting times before commissioning, which isn't possible with larger ships. I also don't want to over stress on the technology, which in recent years, has shown more trouble than its worth, and instead we need to go with proven and incremental improvements in technology rather than large untried leaps.