So if I understand correctly, if I install the main BP component and BP Ascension, I should have Ascension and all the Tougher battles, right ?
Yes
Wait...this looks like a yes and no scenario. If you use the BP core and BP-Ascension, then yes you'll have the tougher battles...and some BP additions If you use "just BP-Ascension", as I hear has been advised in some places--it won't be the same as the former. I know that most of the original Gaider 'tougher battles' is in the actual BP-Ascension component, but don't 100% guarantee that none of it is in the BP core still. Remember, this all used to be one huge mishmash of files mixed together in a cauldron and called sun tea, way back in the olden days.
I can think of one instance off the top of my head; the BP giants and their boulders. I balanced the boulders out in BPv179 cause they were overpowered, but those are a part of BP core that Ascension leans off of if it's there. The boulder-wielding fire giants add a whole new layer of difficulty to the situation, especially if you botch up a save and get your prizefighter stunned for a round or 2.
That's just one example of probably many, and why I recommend using the BP component as well with Ascension-BP. I am trying to work on seperating what is truly essential in BP for the rest of the components, from what is strictly feature--i.e., redesign the core into "essential core" and "general BP enhancements". Maybe even a third component, "BP scripts", seeing as most of you are using a mod that almost completely devours my scripts by the very nature of its installation. I see a few shouts here and there that appear to be BP, but that's about it when I open a BP script. So why bother even putting them in, in the first place? But that's upcoming; for now, yes you can use just BP-Ascension but no it's not the same as BP + BP-Ascension.