There is only one rare house (semi) in Rothwell Heights under 400K.
In New Edinburgh 500K will buy you an apartment.
As far as Centertown core goes (which is a variety of housing) the very top end of your budget would barely even get you a small house to reno. You maybe could find a semi.
Remember that $370 000 is around the average price of a house in Canada. 400K - 550K will not buy you the downtown house you want.
Considering I am in the burbs and my house is 400K (4 bedrooms, 3 baths), detached, with small backyard, 1993 build, I would think you may need to reassess what you are looking for.
Edit: Go to MLS. But in all your criteria and watch a lot of the dots disappear. I for example started with everything in centertown and then put in $550 000 as top price, also changed type to HOUSE. There are less than 10 dots.