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the two times in the comic mizi reject's sua's bid to help her:

first, mizi asks to stay in sua's home and offers to pay sua back for everything by getting a part time job. sua replies that mizi doesn't have to pay her anything, because just knowing mizi is relying on sua and that sua is helping mizi is enough for sua. in reaction, we get the first screenshot.

second, in response to mizi calling herself a beggar, sua says that mizi does not know herself ("why would you say that? you really don't know yourself at all"), insists that mizi is a kind person who hates herself but still tries to be understanding of others, and then says that mizi is being too hard on herself. in reaction, we get the second screenshot.

initial observations: it's interesting that, in mizi's first rejection of sua's offered unconditional help (first screenshot), mizi's vitriol is directed mostly inwards: she calls sua a pushover, but the main focus of her disgust seems to be aimed at herself, as she calls herself a "beggar tak[ing] advantage of [sua]." but in the second screenshot, mizi seems more unhappy with sua as she denies being a "pet" for sua to keep "around the house" (thereby implying that this is how sua is viewing her).

in the first screenshot, mizi's disgust is directed mainly at herself because she is repulsed simultaneously by her own weakness and her exploitation of others; she is a "beggar" with no money for rent and she's "taking advantage" of sua the pushover. if sua had agreed to let mizi pay her back, then mizi would feel more in control of the situation, as delayed payment for housing is an ordinary phenomenon - however, by saying that mizi doesn't have to pay anything, sua takes this sense of control away, and instead makes it so that the sole factor allowing mizi housing is sua "really lik[ing]" mizi. as mizi is someone who copes with being powerless by trying to feel in control and (in alnst) develops a guilt complex about being someone who manipulates others into dying for her, being in this sort of position - simultaneously weak + powerless + reliant on someone's love for her, and taking advantage of someone else's love without giving anything back - would naturally be disgusting for mizi, and would result in her describing herself this way.

then, in the second screenshot, mizi directs irritation at sua on top of the previous disgust at herself. hearing sua reject mizi's self-hating view of herself leads mizi to feel that sua letting mizi stay at her house for free would be the equivalent of sua having "some pet [she] keeps around the house." "pet" is an interesting metaphor to choose (given all of alnst) because it implies at once love as the source of survival (the owner takes care of the pet because the owner loves the pet and finds the pet cute/endearing) and a deeply unequal power dynamic. "pet" is a fitting metaphor because, in alnst, mizi is taken in by and provided for by shine on the sole grounds that shine finds her endearing and so loves her - a similarity, perhaps, to sua saying that she's willing to shelter mizi for free because she loves mizi.

sua said she loved mizi, and then described an idealized image of mizi. thus, since mizi's "i'm not a pet" comment was specifically in reaction to sua giving this idealized image, this implies that (just as in alnst) mizi feels stifled and crushed by sua's rosy vision of her. perhaps mizi uses the pet metaphor because, just as a pet must remain endearing and lovable to ensure its continued survival in its owner's home, sua's words ("you don't know yourself + you're actually kind and warm + you hate yourself") and the situation lead mizi to also feel that, if she were to accept sua's offer of free shelter, mizi would have to keep performing and living up to the rosy, idealized image sua has of her forever - kept around and given shelter purely because sua loves this image of her.

the line "fine. then you can just hit me" then adds to this, because it implies that mizi has made a connection between [sua's offer of free shelter + sua's words] and [the domestic abuse mizi is facing]. if mizi grew up facing domestic abuse, then she probably views being hit as a price she pays for the shelter and food her mother gives her. so, by saying "then you can just hit me," mizi is (from her pov) laying bare the transactional nature of this offer, offering a different exchange more familiar to her (violence for shelter, as opposed to the performance of a rosy image for shelter discussed above that she felt sua wanted), and making obvious the uneven power dynamic that would result from mizi accepting sua's offer.

first, she takes away mizi's independence by inviting her into her home. (to her, relation is enough of a reason - there is no need to repay her if she is considered family).

second, she takes away mizi's identity by projecting the image of innocence onto her. mizi correctly catches on that this reduces her to a pet: a plaything of sua's, lacking agency and dependent on her. again, mizi understands power as physical violence - she offers to just be abused physically rather than mentally as repayment. however, sua want to be hit instead. this way she numbs her own pain, and blurs the boundaries between them physically. this way they become dependent on each other.