I guess you‘d need a MITM SSL proxy to mask your originating IP? That seems to be a bit complicated for this usecase, if Twitter uses certificate pinning?
EDIT: I see now that I had a misconception here. There's no need for a MITM proxy in this case.
Why would you need the MITM part? Just use Tor, VPN or a forwarding proxy to tunnel all your traffic through an IP you can't be connected with.
They (used to?) block most Tor nodes and popular VPNs. But I just found out they launched an onion service last year!