@tggzzISTR that some scopes will only decode what's visible on the screen, not everything that is captured in their memory. That would significantly reduce their usefulness in many situations.
In my experience the three critical features of a LA are clock speed, arm/trigger/filter, and memory depth. Better arm/trigger/filter means the memory doesn't have to be so deep. Ideally the LA's clock input will be driven by the UUT's clock.
If that the case then I will concentrate this budget on a good scope and later acquire a good logic analyzer, or? maybe I can go the hack route save couple of hundreds buck for the LA? That depends on what I end up buying
I am willing to suggest "ways of thinking" and possible alternatives. I am not willing to offer guidance as to what you should do since only you know all your objectives and tradeoffs.
Unless you already have suitable tools and already know how to use them, it usually pays dividends to consider the
details of how you would go about making the measurements with the
simplest possible tools
plus a thoughtful implementation strategy.
Consider, for example, how you might use a very cheap LA ($10-20) which is little more than a clock plus memory. You would probably want to generate a signal saying "this is interesting", capture that plus the data, and then write your own software to post-process the captured data streams to extract only the "interesting" samples.
Or you might have software inside you UUT that blips an output when it detects a significant event, e.g. receipt or generation of a particular message.
In particular, learn to think in terms of FSMs where events cause transitions between states, and code your systems so that there is a direct representation of the states and events. Then use printf() statements and/or blip outputs when events or states or events-when-in-a-single-state occur.
And to annoy rstofer, I'll note my Tek 485 cost £25, but I did have to diagnose and recap the PSU

That's my principal scope on a very small bench, or vertically alongside it. N.B.: I don't recommend you get a broken scope, unless fixing it is an end in itself!