They’ve never been a particularly welcoming place. There are plenty of good bookshops in Edinburgh where one can poke around a real variety of books without being judged on how woke one is.
Their dog is nice, though.
Seriously, I think as a bookshop they have every right to decide who they will and who they won't give a platform to. They're not a public organisation so they can do whatever they believe to be in their commercial interests. That far, I agree with LilaJude.
However, their statement goes beyond simply clarifying their choices (if indeed they needed clarifying). From the statement:
"We do not believe in the zero-sum rhetoric that some of these oppressions must be resisted at the cost of others, or that any particular one is of primary importance."
What they're doing here is deliberately setting up a straw man argument and misrepresenting the position of GC feminists. None of us argue that women's sex-based rights should be of primary importance - we are (e.g.) asking that potential conflicts of rights associated with self-ID are properly identified, debated, and appropriate measures put in place to make sure women continue to be protected. Straw man arguments such as those put forward by Lighthouse add heat but no light, and ridicule and belittle the genuine concerns of many feminists.
"In our view, trans-exclusive rights campaigns, whether they call themselves feminist or use the term “gender-critical”, are reactionary rather than radical, and knowingly cause harm to women."
This is a vile accusation and without evidence. They're entitled to hold that view, but putting it in such a way is likely to suppress engagement with ideas and evidence which I would have thought was contrary to their aims as a radical bookshop.
"As such, people and organisations who share such views with the intention of spreading misinformation that harms trans and non-binary people have absolutely no place in the bookshop or at any of our events."
This has gone beyond no-platforming to 'not welcome here'. They don't define 'spreading misinformation' so presumably it applies to anyone having a conversation on their premises or at their events. I've shopped at Lighthouse and attended their events before - but I won't do so again. That's not just because I don't agree with their stance (I can take disagreement and I'm happy to engage with people I don't agree with) but because they've explicitly told me I have no place there unless I maintain silence on anything to do with sex and gender.
"Many of us have the privilege of not needing to check whether we are safe and included in radical and feminist spaces. This policy is intended as a statement of reassurance to those who unfortunately do."
Actually what you've done is what you said at the start is a bad thing. You've elevated one oppression above others. So maybe TRAs and their supporters feel 'safe and included' in your shop (and arguably did before) but a lot of GC feminists now won't. There are other, and better, ways to promote respectful dialogue. Still, your shop and your choice.
"If any customers have legitimate concerns about this policy, they are welcome to discuss them with us in a constructive fashion, but we will not give time to people who act in bad faith and from a position of prejudice."
Well I have concerns that I consider legitimate but you've been very explicit in your statement that you don't consider them legitimate so I don't think I'll bother. I'm very good at respectful dialogue to try to resolve things when there are disagreements, but I don't think you have the slightest interest in hearing anything that would trouble your world view. For reasons I don't understand, you've decided that large numbers of leftie feminists have suddenly turned into bigots. Until you begin to question the implausibility of such a thing, there's no point talking to you.
"It is not “shutting down the debate” if we refuse to discuss, for example, whether trans women are women; to hold such a debate would itself be immoral."
I don't agree that it would be immoral, but I do agree it would be pointless; at the moment there are two irreconcilable sides who would only shout at each other. Instead, how about holding a debate on how trans individuals should be accommodated in prisons? Or maybe invite someone to discuss the research on ROGD and its implications for how we educate and respond to young people? Or how we respect both the privacy and dignity of transwomen healthcare practitioners whilst retaining women's rights to female care? That might actually get us somewhere. But no, you prefer to paint GC feminists as the baddies so that you can claim the moral high ground. Well I think you might find it's not particularly well populated up there, but I suppose you already know that.