I did a brief glance through Burnerapp's privacy policy and I think you might as well use Google Voice. But I understand some people here simply want to avoid Google.
In the Google Voice settings, you can set "forward calls to" so anyone call your Google Voice number will also ring the forward numbers. Actually in your case you want to keep this empty, so you can use Google Voice separately.
After you create a Google Voice, you can use the Hangouts app for makings call and text. Hangouts is an inclusive all in one communicating app similar to Whatsapp, but better imo because call and text works for actual numbers. One caveat is you need internet connection to use it. Alternatively, you can use the Google Voice app which have finer call and voicemail details, but it doesn't support receiving incoming calls. (To allow incoming call on Hangouts, you have to manually enable an option in the Settings -> Phone number -> Incoming phone calls". You can also enable "Upload contacts" if you want the app to read your iPhone's contacts list.)
The other thing you can do is use RedPocket prepaid, they have cheap plans.