It will really come down to where you want to be working and how selective you want to be with choosing which PE fund. Assuming you are referring to a deal team role, the good news is that there is a longtail of PE funds you can consider (e.g., mid cap energy fund) if you're open minded about the brand, but they will still be super selective (tier 2 to PE/VC is not common) and will usually hire locally.
1. If you are looking to change geographies then an MBA will make a big differrence. As you already know, PE/VC is super competitive and local where many like hiring top performers from top MBAs, who naturally tend to have a strong pre-MBA background as well
2. The general thesis is that you need to learn the PE toolkit asap from scratch which is partly why they like to hire early professionals (2-3 years of experience) but they typically say that a consultants ability to analyse a business holisitically (i.e., business DD) is a transferrable skill
3. I have interviewed at two of the firms you've listed where they all have a strong bias towards bankers over consultants. The interview is a mix of personal fit, technical questions, deal experience, investment thesis, LBO modelling test and sometimes a investment case study. The number of interviews can really vary where some will expect you to meet with the majority of people from that team across levels and all partners/MDs
4. For a consultant, it's really the accounting and finance basics behind an LBO model and ideally trying to get as much DD experience as possible so that you are good at developing an investment thesis
5. Much of the recruiting happens through headhunters so it's super important that they know you and see you as a strong candidate. Referrals do happen but are not super common unlike consulting and so I would focus more on the network building. The networking is most naturally done through alums of your company who have joined that fund - if there aren't any that's usually a sign...
6. Unless you are a US citizen or have prior experience in the US, large cap PE funds will not consider you as they already have a wealth of talent from IBD, MBB and top MBAs to pick and choose from