If you don't have time to meet the requirements, then you didn't give yourself enough time. If you are limited by time it's presumably because you are working for someone else, and you're constrained by their timetable and/or budget, so you need to build on what you know.
You should never blindly implement on a framework you don't know. You need to build time for prototyping, learn on your own, or pull in talent that already knows. If you are using a new framework and there is uncertainty, you should have enough time that you can cut after a certain point and rebuild in a way you do know you can do it.
You should also build on the best framework for the job -- there is no one-size-fits-all solution for any problem. What framework you choose should be based on the user requirements, technical/business limitations, projected growth and life (maintenance needs), and your available resources. Choosing is based on your personal past experience, and prototyping. If you don't have the experience and don't have time to prototype in at least a couple frameworks, you are in a bad spot right from the start.
That all said, I personally think ending up with a "well designed" system in WebForms is INCREDIBLY difficult, especially compared to doing it within MVC, and that in the ASP.NET space MVC is a much better framework in almost all situations than WebForms. I am biased and will outright say I think Webforms is fundamentally flawed in its philosophy; most of the code works as small examples but falls apart and becomes unmaintable as you grow, and even the example code from MS suffers this issue. But you should not use some random person on the internet's opinion as the basis for your design choices. I am just pointing this out because I want to be clear that I am not endorsing specifically WebForms, I am endorsing "use what you know" -- whether you "know" it from previously using or take time to learn it.