Best Chatbot Service for Website Integration for Research Purpose (2021)

I am a researcher and I do conduct research on conversational agents, chatbots, anthropomorphism and human-computer interaction.

For a series of online experiments I need to implement a functioning chat. I already conduct a few online experiments with a dummy chatbot to measure the mere presence of conversational user interfaces.

Now I am looking for a functioning chatbot so that my participants can interact with the chatbot. I was already looking into Dialogflow, BotFramework and various other services. However, I do have some requirements

1. The chatbot should be integrated into a website. The website already exists and is developed using plain HTML,PHP,JS.
2. The chatbot should be able to take data from the website (i.e. user_ID, treatment condition etc.) and should be able to adapt accordingly (language, design, features).
3. The website should be able to access the chatbot conversation and save it into a DB (I'm using a simple MySQL)

Any recommendations?

Currently I want to use DialogFlow and the Dialogflow Messanger, which however only has limited styling options (change of color etc.). Is there any saas for integrating the chatbot on the website?

Also keep in mind, in research, we unfortunately don't have much funding :D

Thanks Dominik