What is Haywood’s view on patriarchy in light of Fantomina’s and Beauplaisir’s experiences? Is she critical of the uneven consequences of characters’ actions?

Eliza Haywood's Fantomina seeks to expose the double standards of 1700s society, particularly in regard to gender, and therefore is critical of the patriarchy.
Fantomina is a young woman who tricks her lover Beauplaisir by using various disguises to seduce him. While Fantomina (also one of her disguises, by the way) is able to enjoy sexual pleasure for a while without explicit consequences, she eventually becomes pregnant and is sent away to a monastery as a punishment for her behavior. Her lover, however, faces no punishment. The ending of the story shows how men are privileged in a patriarchal society and how women are, by nature of their biology, subject to the burden if an unwanted pregnancy occurs.
He is also able to carry on these affairs with supposedly different with no sense that he's actually sleeping with the same woman. While Fantomina herself devotes lots of thought and energy to her game, Beauplaisir is clueless, showing how he is detached from the personal aspect of sexual interaction. While Fantomina cleverly plans her disguises and gets into character, Beauplaisir moves from woman to woman without a second thought.
The characters' respective behavior throughout the story and vastly different consequences at the end of the story serve as a critique of gendered double standards and by extension of patriarchy at large.

Haywood presents a critical view of patriarchal society in Fantomina. In the story, female identity and individuality are rendered unviable by patriarchy. The fact that the protagonist is nameless, only known under the names of her various disguises, makes her a nonentity, powerless and restricted by society. She is introduced in the very first sentence as “a young Lady of distinguished Birth, Beauty, Wit, and Spirit.” Other than this generic phrasing, she is given no truly distinguishing characteristics. Fantomina, the name of her first disguise, creates for herself a number of differing identities in order to pursue the love of Beauplaisir, a man who does not truly care for any one of those disguises (nor the lady herself).
There is also the point that Beauplaisir does not recognize that the different women he is seeing are in fact all the same person. He does not know that each successive disguise is actually the protagonist until she confesses her masquerade to him. He does not truly see her because in their society, the protagonist does not have a specific self—she is a female role rather than a self-actualized person. This generic female self is what enables her scheme to work in the first place.
While her series of disguises seems like a form of power to the protagonist, they are ultimately a lost cause. She does indeed have a form of control and she does effectively manipulate Beauplaisir, but this fruitless. She undertakes all of these actions that further lessen her sense of identity for a man who does not care for her. He does not end up loving her in any one of her personas. Thus she forces herself to take on these disparate roles in order to keep his affection and she loses her freedom in the process. Beauplaisir, on the other hand, maintains his freedom and his identity throughout the story.
As for the consequences, the protagonist ends up pregnant and without the man she loves. Her mother sends her to a monastery in France as soon she is able to do so. This would have been a sobering life at the time, thwarting her from any possibility of personal freedom. As for Beauplaisir, he faces no lasting consequences. While he offers to take charge of the child, he does not offer marriage or any other form of commitment to the protagonist. When denied the child, he “took his Leave, full of Cogitations, more confus'd than ever he had known in his whole Life.” Beauplaisir simply leaves confused and perhaps a bit troubled, but certainly no worse for wear. Haywood presents this disparity in consequences as a matter of fact. It was how things were at the time, but by presenting a sympathetic figure in her nameless protagonist, she offers a subtle critique of the patriarchal society that engenders such stark realities for women.