I don't think the welfare state is particularly relevant to the policing of what is acceptable 'feminist though' in the context of PP.
PP is, as noted, 'privileged', in that she has the means to do & say what she likes with no fear about job loss or whatever.
This 'privilege' (I use 'privilege' in scare quotes because 'privilege' is something that is frequently asserted when perhaps it isn't the main factor at work) is a factor of opportunity (she is comfortably off) but also choice (she chooses to put her head above the parapet).
In general terms certain points of view are risk free to hold in the 'corridors of power' (politics, media, education), and they include TWAW & related thought. That doesn't make holding 'non-BBC' opinions of itself 'stunning & brave' - there's nothing terribly interesting or novel about thinking that homosexuality is immoral, or whatever else. What it does mean is that opinion that contradicts an orthodoxy is dangerous if you express it to an audience that firmly believes that certain views are bigotry, and that bigotry is the worst possible sin to hold, and where your 'bigotry' is subject to policing via the same.
For example, a Muslim Labour councillor agreed with a Muslim mother that a primary school curriculum that teaches about LGBT was inappropriate. www.birminghammail.co.uk/news/midlands-news/muslim-councillor-probe-after-endorsing-15747253 Looking at the two parties here, the Muslim mother is subject to relatively little 'thought policing', in that she does not hold a position of power, and because her religion is a minority one, so her religious views are given a greater degree of leeway than if she were not a Muslim. The Muslim councillor shares her religion, and had initially shared her viewpoint, but he was 'thought policed' by his political party, which said that his views were contrary to those of the Labour party, though they are plainly uncontroversial for those of his religion.
As far as PP goes, she is afforded less leeway because of 'privilege': it is seen that she is white & comfortably off, and as such her thoughts are ruthlessly policed when they contravene any aspect of conventional liberal opinion in any way.
As to WHY this is so, it comes back to bigotry being the worst possible sin for the liberal establishment, but also in some way to a desire to construct a simple victim/oppressor narrative. Transwomen are often an 'othered' group: if we look at the kathoey of Thailand, they are not the ruling structure, nor are the transwomen of Brazil, India, etc. So the lazy victim/oppressor narrative says that transwomen are a victim group and so anyone who opposes them is an oppressor.
This narrative persists because it is convenient - people aren't interested in pursuing the nuance of argument, they just want a simple solution. So when you have PP who is white, fairly wealthy, etc., well she's the perfect villain isn't she.
As for who her victims are, well we could perceive the likes of Jane Fae, Munroe Bergdorf, India Willoughby etc. as very middle class and privileged - and of course they are, but that's not the dominant narrative, and it's certainly not an easy sell. It's much easier to sell 'transwomen are women suffering in the 'wrong body' facing bigotry from nasty rich, white oppressors like Posie' than it is to deconstruct that narrative and look for the nuance. I mean really who wants to do that? No doubt transwomen suffer discrimination, and for the the liberal well-meaning establishment and their adherents, that's sufficient to come up with the simplistic conclusion that works well enough: for anyone who isn't either some sort of right-wing bigot, or a rare person who has time in their life to look into this in lots of detail (and who does, honestly).
Hence we have PP as the rich white oppressor and the TW as the victim group. And of course because PP is of that class and she's not embarrassed about her opinions then she will be written off extremely quickly as Donald Trump/Nigel Farage. The same fate would not occur to the Muslim mother I mentioned earlier because she's not of the dominant class.
Any power that Posie possess through money is of course nothing compared to the greater physical strength of a typical transwoman compared to her, but that's not really the issue here - Rupert Murdoch might, at 87, be physically weaker than the average woman, but he has much more power.
So Posie's arguments about the real threat to women posed by male-bodied individuals are problematic in that sense also, because they are more likely to be seen as attempts to use her obvious power to further oppress an out group (TW) than they are to be seen as a legitimate attempt to protect women & girls.
Of course the problem is that while PP has some power to stand up and make a bit of noise, it's not real power that actually gets things done. If you own a tobacco company that sells $10 billion of tobacco that's real hard power. Posie's power is simply the power to open her mouth - it's not the power to be listened to and respected.
(And let's not lose sight of the fact that her self identity as a 'wife' has implications: the Quran (4:34) states "Men are in charge of women, because Allah hath made the one of them to excel the other, and because they spend of their property (for the support of women)", and that's hardly a view specific to Islam, 1 Corinthians 14:34 holds "Women should remain silent in the churches. They are not allowed to speak, but must be in submission, as the law says."
The connotations of 'wife' are as chattel, maintained by the husband, without independent thought.
And alternative are scarcely better - the dreaded 'single mother' is 'paid for by my taxes', and again has no right to hold any opinions as she 'is not paying'.)