Benefits adviser here. I don't think your MP will be of any meaningful help. You need a benefits specialist and for your daughter, one of the autitistic charities may well have someone who can help.
The PIP decision making process is surprisingly technical. Underpinning the questions on the form is a vast amount of regulations and case law.
First, take a look here www.citizensadvice.org.uk/Global/Migrated_Documents/adviceguide/pip-9-table-of-activities-descriptors-and-points.pdf
This is your starting point. This will enable you to work out how many points your daughter should have been awarded.
When looking at each descriptor, bear in mind that if someone cannot carry out the activity safely, repeatedly, to an acceptable standard and in a reasonable amount of time. If the applicant cannot carry out the activity to those criteria, then they can't do it. So, for example, if your daughter takes a long time to dress, because she can't decide what to wear, or choose appropriate clothing for the weather or circumstances, then she can't dress without support, help or prompting and points should be awarded.
Prompting and encouragement count as needing help from another person.
The inability to carry out an activity must be related to a medical condition or some other problem. Eg, if someone has no motivation to wash or bathe because of their depression, they need prompting/encouraging and should get 2 points. If they are easily distracted because of ADHD or something, they may be inclined to leave food in the oven or on the hob, so they cannot cook safely without supervision, and should be awarded 4 points.
So you need to start her mandatory reconsideration. Go through each descriptor in turn and work out what points you think she should have been given, and why.
List them, in order, and explain 1) what issue makes it difficult for her to do the activity, what sort of help she needs to do it safely/repeatedly etc, and how this is addressed, eg you supervise or encourage/remind her.
Re the preparing food/eating, the threshold for preparing food is a proper meal prepared from scratch using fresh ingredients, so a bowl of cereal or popping a ready meal in a microwave doesn't cut it. It's not an "acceptable standard" nutritionally so doesn't count for eating - spell that out. (I'm surprised by that, I've never seen them make such a schoolboy error lol)
The "worst day" is not a factor in PIP, that was DLA. The test in PIP is "the majority of the time" so if the help is needed every time, say so. If it's less, say so; eg "once or twice a week, X manages activity Y on her own, but most of the time I have to (prompt/supervise/assist - whichever is relevant) ... and explain the help she needs.
Work out what points you think she should get for each descriptor and tell them, eg "Therefore descriptor 2d applies and 4 points should be awarded".
Whack the mandatory reconsideration in. If you don't have time to do a full job, write a letter for her sign asking for a mandatory reconsideration and saying that grounds for reconsideration will follow.
Supporting evidence from professionals is always helpful. Anything to do with her diagnoses will help, but if she's involved with any other services, attends any events connected with her autism, have any reports (these could be from when she was at school, as autism tends not to change much, and I have successfully used an EHCP before now) these will help and can all be sent in later.
There is also some case law about "exceptional motivators". I had a case where the DWP dismissed my submission for some descriptors because the client went horse riding once a week. This was not upheld because I submitted that this was an "exceptional motivator", ie her love of horse riding overcame her difficulties.
Any factors that may have disadvantaged her at assessment should be cited. I assume it was done over the phone, so it may be that she manages telephone conversation more easily than face to face, and that they will not have been able to observe physical signs of how distressing the experience was, but she was actually shaking, hair-pulling etc. They could not form an accurate opinion if they couldn't see her.
Sadly, the mandatory reconsideration rarely changes anything. I'm convinced it's just there to try and reduce the number of people going to appeal. The only decisions that seem to get changed at this stage are those where the client missed out by a couple of points. But it has to be done before an appeal can be submitted.
When it comes to the appeal, your grounds for appeal can be essentially the same as those for the MR, so you'll already have done the hard bit and can just copy and paste what you've already done with a bit of tweaking.
Don't be discouraged. The success rate for appeals is incredibly high, over 80%. (This rises massively when benefit specialists are involved, and 3 out of 4 on my team have 100% success rate). I'm confident that your daughter would win at appeal.
And when you win, the awarded is backdated to the date that the PIP would have started. Some clients have received 5-figure sums in arrears payments.
The system is cruel and unfair imo, but this is what people voted for when they voted for the party that were going to reform the "broken" benefit system. They reformed it by smashing it to bits imo.
Good luck.