The lesbian forums on Reddit, but for one, are all moderated by transwomen. Autostraddle, which began as a site for lesbians, is now run by transwomen. Almost all feminist sites are owned by men. After Ellen, too, began as a site for lesbians and was taken over by transwomen, but at last that site has been purchased by a lesbian who is again publishing it for lesbians and has a great deal to say about the problem posed by transbians in lesbian spaces, for which their increasing readership is grateful.
There is virtually NO lesbian representation, and certainly very little spending of monies for lesbians, in any of the major LGBT organisations. In fact, lesbians are being thrown out of leadership positions in such organisations — see for instance Martina Navritolova and Julia Beck.
So I would suggest that the fact that lesbians are being silenced and their voices being sock-puppeted online by transactivists means that you cannot gauge lesbian support for transactivism by what you read online.
A more interesting gauge is the recent study done showing that something like only 9% of lesbians would consider a transwoman as a sexual partner. That tells me that 91% of lesbians do not believe TW are actually women.
I’m quite sure that most lesbians, as is true of most feminists, want TW protected from discrimination in employment and housing and want TW safe from male violence. It’s possible to legislate for that without creating laws that say TWAW; it’s possible to protect TW as TW while also protecting women as women.
It is the insistence that TW ARE women in each and every respect and therefore entitled to everything women have fought for to alleviate millennia of subjugation TW were never subject to that is the sticking point.
That, and the issue of forcing people to believe something counter to material reality.