It's ten years since I adopted, but a significant proportion of my prep group brought their own lunch to the sessions rather than eat the buffet provided because they were engaging to a greater or lesser extent in weight loss.
What I am about to say is probably going to make for uncomfortable reading, but I think you need to hear it if you are serious about pursuing adoption. The child's needs have to be at the heart of a child being placed for adoption. At the moment you are focusing on yourself and your partner, and that is absolutely fine for the stage you are at, but over the coming months (or perhaps years if that's what it takes) as you go on this journey, you will need to shift your thinking to what does this mean for a real (not hypothetical) child.
You've said that your weight/mental health would not get in the way of the love and care you can give to a child. On a surface level that might be true, but many or even most, adopted children have some level of additional need, which may not be evident when they are placed. Love and care are really just the tip of the iceberg of what they need.
All children need a certain amount of energy to 'keep up' with them. I'm currently nursing wounds from thinking I could cycle up a hill to keep up with my eleven year old. Adopted children often have an extra level of challenge in this respect. They will test you mentally and physically in ways you will not have experienced before. You may have a hyperactive child who really needs to get out to the park to blow off some steam to regulate, but you've been up with them in the night due to night terrors or bedwetting, or a child who runs away from you whenever they feel anxious, even if you're in a busy shopping centre or at a train station. Adopted children may have difficulties socialising with their peers, or just don't fit in due to being a bit different or presenting as younger than their chronological age. Care givers of adopted children are more likely to need to actively engage in their children's play to plug that gap when friends are in short supply. None of this is hyperbole. These examples are all based on adopted children I know.
I am saying this to prepare you for what could be coming your way. There are guidelines for adoption agencies regarding bmi (you can google this). This doesn't mean that you will be prevented from adopting based solely on having a high bmi. Others above have said it's possible. But you can see from the examples above why it makes sense for prospective adopters to be in the best health they can be, and to want to make changes in order to be the best effective parents they can be, not only right now, but ongoing for the next 10-20 years. As mentioned upthread, it's as much about making that commitment to wanting to improve your health, and being able to take action on feedback given, than the actual BMI at the end of it.
Consider as well, the staff shortages mentioned above. If the agency only has resource to progress with half the people that apply, they will want to go forward with those who the feel are most likely to go the distance in terms of the approval process, having a child placed and the all-important parenting that comes at the end of it all. That might feel unfair, until you remember that adoption is about getting it right for children, and not about the needs or desires of prospective adopters.
The 'obstacles' being thrown in your way are certainly not unusual, and most adopters will have their own unique story to tell about this. At times it feels like you are being tested for your patience, resilience and self-advocacy. Often delays though are purely down to lack of resource. Honestly, the challenges of getting through the adoption process have paled in comparison to the challenges I've faced since my child was placed. You are probably aware of the SEND crisis - and if not I would suggest you read up on this fairly urgently. Children are waiting years for autism and adhd diagnoses (and then add on at least another year if your child needs ADHD medication), years for ehcp plans which often involve a lengthy fight and a sometimes a tribunal, years out of school waiting for a suitable school place to become available. You might have in mind that you would not want a child with high needs, but the fact is that it's the needs of children that sit somewhere in between high and low (where a lot of adopted child sit, in fact) that are falling between the cracks. If you adopt a child under 4, their level of SEN likely won't be apparent. If you adopt a child 4 and over you will at least have a better idea, but many children do not get a formal diagnosis until they have been at school for a couple of years.
The delays I faced waiting to adopt, and there were many, were honestly a walk in the park - quite literally when I could use that time to work on myself, my house, take a holiday etc. Delays when you have a child out of school and you are trying to hold down a job are next level in terms of testing you. Adopting my child remains the best thing I have ever done, but it has been very challenging on many levels.
I know that won't have been an easy read. It was not meant to put you off, but you do need to at least take a step back and look at the bigger picture to make sure this is definitely right for you. I hope it has given you some food for thought. Good luck!