Actually not sure what a tri shoe is. I guess there are some tri-specific models of both run and bike shoes, but not entirely sure what the difference is. Maybe suitable to be worn without socks? I can't do that anyway due to a bone spur on one foot. Even a sprint can be painful without socks, and I mostly do longer races, so it's not an option. Any running shoes can be made more tri-friendly by adding Lace-Locks or similar so you don't have to tie shoes in transition. I think unless you're really committed to sockless racing, or doing most of your races in super-humid, wet conditions where drainage is a big deal, there's no need to look for tri-specific shoes, just whatever is comfortable for bike and run.
Bike shoes in my experience tend to be expensive. Due to foot issues I tend to end up spending more $$ than I want to to get something that really feels comfortable. Good bargains can usually be found online for running shoes once you know what works for your body. Definitely no need to spend $150. Most of mine are in the $50-$100 range, depending on model.
(If you can find remaining shoes from a previous year's model, tends to be cheaper.
)
ETA Actually my newer bike shoes ARE tri-specific; did not know that when I bought them. I had Sidi's for many years--great fit and quality but pricey and they were leather, which shrunk over time after getting wet a lot. My newer ones are Specialized TriVent SC. I've had no trouble with them--they are amenable to different inserts
(Specialized makes a series of these which are compatible with their shoes if you don't like the stock ones
), have a roomier toe box than a lot of bike shoes, and are easily adjustable. Quick to put on/take off with Velcro straps. I think they were under $150. As I recall, Specialized has a higher-end tri-specific model which uses a kind of dial to adjust the tightness--super-fast to get on and off and adjust on the fly, and even more comfortable. They were over $200 at the time
(but this was in Vietnam, where import tax can be really high, so maybe cheaper elsewhere
), and I decided the cheaper ones would serve!