Help! Estranged Mom asked to move in with half-sister!

I (26f) have a half sister that is ten years old. She is my mom’s child. We live states away from each other and I’ve met my half sister maybe five times in her entire life. My mom and I never had a great relationship. She and my dad divorced when I was 12 and my dad had primary custody. She got me a few weeks out of the entire year and some holidays until I was 17.

I am now married to my wonderful husband and we have a 3 year old daughter. Over the past year, my mom has called me only a few times, typically months in between. (This is the norm of our relationship, she missed my prom, high school and college graduations, and nearly my wedding). The conversation often is along the lines of “I am depressed, we can’t afford anything, we’re going to be homeless, I can’t buy groceries.” Typically, each of these calls results in her asking for money (and me sending it). Not once has she called to check in on me, or her granddaughter. She has only met her granddaughter one time and that’s because I went to her.

Fast forward to today, a couple of days after my birthday, she calls me to tell me happy birthday and then follows it up with asking if her and my half-sister can move in with me. I’m angry, upset, and overwhelmed. Immediately, I told her no, that it wasn’t an option. 1. I don’t necessarily have the room in our house, 2. My half sister is allergic to cats, and we have six. 3. My mom has caused issues in the past, and I don’t want that around my daughter.

She told me they would have to move states to live with an old friend or she would have to surrender my sister to the state.

Immediately following the conversation, I spoke to my husband and he was furious because he feels she always manipulates me emotionally. We have helped her in the past, including driving 24 hours to get her and my sister five years ago to escape a her bf/baby daddy because he physically put his hands on her. She also, unbeknownst to me at the time, had a DV on her record against him.

We offered to her then for them to stay with us and her turn her life around here, but a week later she demanded we take her back.

We’ve decided we will absolutely not let her move in with us, because who knows when she would actually ever leave (she has two evictions in the last two years). However, we don’t know what to do about my half-sister. One, the allergy to cats makes it next to impossible for her. And our animals have been with us through a lot (miscarriages, deaths, etc.). We will not be getting rid of them. She’s also practically a stranger to us. And is moving across the country the right thing for her? I also fear that it will disrupt our household and take away time from our child and even put a wedge in my marriage.

Honestly, I’m feeling lost and don’t know what to do or even how to feel. Anyone have any advice?