I'm sorry you had to deal with that this evening

Never an excuse for rudeness. And not to excuse poor training, but unfortunately, phone CMs are not always the most knowledgeable about what happens in the parks -- the call center may not even be located in FL and some phone CMs may never have set foot in the parks.
However, in this case the CM is right in some respects. Determination of
DAS eligibility can only be determined by Guest Relations CMs once you arrive at a park. Nobody on the DIS can guarantee a DAS, nobody on the phone or email can guarantee a DAS.
DAS is for hidden (non-visible disabilities). It is not available to those with strictly a mobility or stamina concern. Based on what you posted above, your daughter would be accommodated by a stroller-as-wheelchair and likely not qualify for a DAS. If she has other needs that impact her ability to wait in a standard queue environment, you should be prepared to discuss the NEEDS and not the diagnosis (or equipment, etc.). A DAS will not eliminate the needs to wait, just allows you to wait elsewhere while your place is virtually held in line for you. DAS entry is through the FP+ line, which at busy times has been report at upwards of 20 minutes.
If heat/humidity is a problem, you might want to consider a trip during off-peak cooler weather. If that is not possible, you might look into cooling vests or such. Maybe a misting fan. Stay hydrated.
I believe I have directed you to the WDW DAS sticky near the top of this forum in another post. If you have specific questions, feel free to ask and people here will try to provide insight to help you plan.
Enjoy your vacation!