If you're still struggling with disordered eating, I would really like to encourage you to find some sort of therapy. It's a very hard thing to overcome without support, and it's a very difficult way to go through life.
That said, how much does your weight go up after an entire week of eating normally? If you are used to eating very small amounts of calories, you are probably also eating small amounts in weight of food (not counting water content of food). When you eat normally, all of that food weight does not leave you by the next day, and your body is also probably retaining water to help you digest the food. This could easily be why the scale jumps up a few pounds the next day! It seems very unlikely that you could put on a substantial amount of body weight or fat in a 24 hour period. Maybe you should see how your weight changes after a whole week of eating normally. If over the course of the week you've only put on those first few pounds, it's probably just the food in your system and not actual weight gain.
I am not a doctor, so I am just speaking from my own experience. I am vegan, eat a lot of fiber, and have a very consistent elimination process. Even so, if I don't eat solid foods for a few days, I drop weight quickly but when I go back to them, I gain a lot back right away, even if my elimination system appears to be working as normal. For me, most weight fluctuation that's measurable on my scale within a 24 hour period is really just a reflection on the change of water and food weight within my system, and not an indication of a change in my body fat.
Looking at your signature, it seems we've had very similar weight histories. I'm also 5'4 and my heaviest weight was 142, while my lowest was around where your final goal weight is (disordered eating has been in my life as well). Now, I don't know how your frame differs from mine, but I do know that not a single person thought I looked healthy at my lowest weight. I didn't feel healthy, and I still thought I should lose more. Please be careful with your goals! Right now, I'm at 120 pounds and my goal is 110. I know I have to get there slowly and not get tempted to go lower because experience tells me that that's how I start to spiral out of control. If you need any healthy support here, I'd be more than happy to try.
Please try to find a way that you are comfortable being healthy!