How do we fight?
Parliamentary
Any bills will be tabled and debated in Commons with the possibility of amendments. We have a core of about 10 or more potential Labour rebels if the bills are bad enough. That is not enough to stop the bill, but is enough to cause a media storm if they rebel. The Tories will go all out because what do they have to lose. It is not the vote which they will lose, but the weaknesses they will expose.
You mention lib Dems/Greens etc. They are not certain to support the bills. Fact is they won't because the Bills will have some stuff to keep women happy. So they will be trying to amend them to make them more crazy. TRA Labour MPs will be tempted to support them. More delays and debate. More sunlight where Lib Dem MPs will have to argue for removing safeguards etc. They may back the bills in the end, but the process will be messy.
Then there are the four Gaza Independent MPs. They are only 4 votes, but I think they will not be keen on these bills as they might lead to Mosques being in trouble over conversion therapy, or saying women exist and they need to be dressed in religious garb etc. So that will lead to infighting with TRA leftists/progressives which could get messy real fast. Again more debate. Do Labour want to lose Muslims over gender stuff?
Then the bills go through the Lords multiple times also with possibility of amendment. Any weaknesses will be mercilessly exposed in the Lords. The Lords might very well vote them down. Which means more Commons votes.
It will not be a straightforward process. It will be heavily contested.
Media
Most newspapers will watch gender bills like a hawk. Multiple stories will be run.
Even if the BBC skimps on coverage the social media buzz will be huge will get a lot of interest. Above all if JKR tweets it is a news story. That will go on for months.
Protest
We have a LOT of women, even men, ready to act and protest in person and online. You think JKR's army is big now well I think it could get way way bigger if we can argue the existence of women is at stake. Lots of demos outside Parliament.
So you might say all this doesn't stop the bills being passed with that majority and Lib/Green support. Nope, it doesn't but everything we do raises the stakes and effort needed and the attention it consumes and the drama. That is important. That might mean the bills are weakened or amended due to popular pressure.
We follow the Women won't Weesht playbook. They had way less resources, less media attention than a Westminster debate would have. The bill also did not have to face the Lords. The bill was passed, but the SNP was critically wounded to do it.
Sturgeon went all in to get the most TRA friendly bill possible to pass. Remember SHE LOST. It all was wrecked in the end.
How much effort will Starmer put in? 80%, 60% 30%? The thing is how important is this to him and the cabinet? I don't think he cares that much.
Then we get to the next fightback
Legal
We have amazing lawyers. They can analyse the bills and point out errors. Say Labour pass something which makes no legal sense. We challenge that in the courts and keep on going. We have lawyers and we have the funds. The clever in Labour know this. So they will be keen to avoid judicial review by drafting a bill which isn't open to challenge. However to do that they need to concede a lot of ground to reality. TRAs will not like that at all. Which makes the Parliamentary debate more fraught.
Which leads to my final point.
Operation Let Them Speak
Our biggest allies will be the TRAs themselves. They will describe any bill which admits reality as genocide. They will undermine the more dangerous clever activists who cloak their aims in favour of the most crazy ones. They will make Fox Killer their legal expert. Right through this process Labour will be being described as monsters. Which links back into the question about effort.
How important is this stuff if you are going to be described as transphobic bigots whatever you do anyway?
Knowing if you screw it up a child rapist might be waving his GRC weeks after the bill is passed saying "thanks Starmer".
That you might have child psychologists being prosecuted for telling girls they are not boys.
Yes bills are likely to be passed, but we can challenge them and hope to amend them enough they are dead letters. Or make the bills so toxic through publicity they badly damage Labour electorally.
Plus ultimately a Tory/Reform government would tear them up in 5 or 10 years anyway if they anger people enough.