Remove direct bitcoin calls from qt/paymentserver.cpp

Summary:
Partial backport of Core PR 10244
https://github.com/bitcoin/bitcoin/pull/10244/commits/3cab2ce5f9e159ad5a2e9ed682f28121b5248580
Progress towards T417
Depends on D2885

Test Plan:
make check
ninja check
bitcoin-qt # sanity checks

Reviewers: deadalnix, Fabien, #bitcoin_abc

Reviewed By: Fabien, #bitcoin_abc

Differential Revision: https://reviews.bitcoinabc.org/D2886