The reality is that maybe a decade ago, it was LBS (or INSEAD) or bust. There are a handful of schools in Europe that have become much more competitive in that time, you can see them here:
The Improving State of Management Research in Europe (An Academic Post)
You still maybe only have 5-10 schools in Europe that are across the board very very solid. You have maybe another 5-10 schools are a bit more "specialized," they may offer excellent training/mentorship for some specific fields and for others they are not as strong.
Cass will not necessarily work against you, but it is not a high status type school that will get you a fly-out based on the name alone. So if you want to work in the US, your currency is going to be two things: 1) your network (which is really a combination of your own network but also your advisor) and 2) publications. In the US, 5 years is a normal length of time for a PhD, I am assuming Cass is 4? So immediately, you need to be 1 year ahead of your counterparts in the US. That means thinking about projects from day 1, learning the academic language fast and pushing yourself to get pubs out in your 2nd/3rd year. It is not easy. Of course this is not the only option, you can consider post-docs in the US to build a network/give yourself more time, or thinking about beyond your first placement how you could keep publishing / a pipeline that would be enough to go to the US.
Overall, I don't think it is impossible. The typical advice you will hear is that it is much more difficult to move to the US from Europe. I don't think it is as bad as it used to be. Training is much better, the schools are much better, and so there is awareness that they do quality work. It will not be an easy road, but there are also several types of schools in the US, this advice is mainly at research roles (e.g., R1 schools). The challenge, to me, is how to do 5 years of work in 4. If you have a good project/advisor from the beginning you will be ahead of the US where you don't have this model.