I think that you are thinking about it in a very roundabout way. It seems that your underlying assumption is that homophobia and all other phobias associated with gender and reproduction are the norms. Therefore, you need to find a way to fight those phobias to make society more accepting and tolerant toward LGBTQ+. However, if your society is tolerant from the beginning those phobias will not exist at all. And your goddess and her priests will have no need to spread the messages of love and tolerance.
I would suggest building your society on these principles:
- sex is disassociated from reproduction;
- affections and sexual attraction have a value of their own that is higher than the value of reproduction;
- gender is seen as fluid;
- LGBTQ+ behaviour is more common or more visible compared to our society.
You will most likely end up with promiscuous behaviour as a norm unless your species are true monogamous species.
(1) Sex: primarily pleasure and social bonding, reproduction is secondary
Your species should engage in sex as an activity that is not necessarily related to reproduction. This is not a farfetched idea, various monkey species do it. If reproduction is secondary, mating with a 'properly' sexed partner is much less important.
If reproduction is secondary and sex is primarily used for social bonding and pleasure, the appearance of religions similar to Abrahamic religions becomes unlikely. Abrahamic religions equate sex with reproduction in their orthodox forms. Thus, they create rules that are meant to increase reproductive success (for both males and females).
Era will need to pay attention to any early signs of religions that focus on reproduction and attempt to control sexual behaviours. Kill all prophets if necessary.
(2) Affection and sexual attraction are needed for reproduction, but reproduction is not the ultimate goal when it comes to relationships between mating partners
The first part is necessary to make sexual relationships consensual and prevent rape. The second part aims to place feelings above reproduction and give freedom in choosing one's partner. If this policy is implemented to the fullest extent you should expect promiscuity. You can try to counter it with social monogamy, but you need to remember that social monogamy has a goal of increasing reproductive success and links sex with reproduction which is counter to (1).
Era can promote the value and beauty of love and affection. She should also promote the idea that sex and gender are irrelevant. Love is strange and can create all kinds of relationships (or pairings if you need couples). You can try to introduce the notion of romantic love early, however, it is a big can of worms that I would suggest studying before opening. Western notion of romantic love is not as pure and innocent as it looks. It surely does not take into account that affections change and that sexual attraction and romantic feelings do not always have the same object.
(3) Gender fluidity
The notion of gender fluidity is necessary for the acceptance of people who do not identify with predominant sex and gender roles. This can be achieved in many ways. For example, at the early stages, the idea of souls can be very productive. If souls are independent of the bodies it is easy to explain and accept that sometimes souls end up in bodies that do not match their gender. Blame it on a drunken minor god who was in charge of distributing souls. Once your society learns about genetics and studies the brain sufficiently well, they will naturally find other explanations for gender fluidity (just like we are finding them now).
(4) Make LGBTQ+ behaviour more common
There is no real need to make everyone homosexual/pansexual/etc. Just increasing the number of LGBTQ+ individuals in such a way that every community has at least 1-2 of them should be enough to make the majority of people somewhat familiar with them. If your species are similar to humans, familiarity would decrease the potential for fear and hostility. Humans fear the unknown the most. And humans are very hostile toward things they fear.
I would expect that just implementation of (1), (2), (3) would lead to a significant increase in visibility of LGBTQ+ because no one will need to stay in a closet. So, you might not need to tweak any numbers.
I want to mention one more thing. Matriarchy will not play a significant role when it comes to sex and gender. Women enforce social norms and rules no less than men do, even if these are patriarchal norms and rules that impose restrictions on women.