Reverse DNS required to send emails

Typically the reverse DNS for the IP is managed by the ISP. This is not strictly speaking a guarantee, but I would be highly surprised if you were actually in charge of the reverse DNS record in this case. Generally speaking, a company purchases the use of those IPs from the ISP and as such, they maintain all IP specific records, including reverse DNS. Basically, whoever looks at your BIND database for DNS will see the record fine but assuming that is not the public authoritative DNS server for your domain setting the record there does you little to no good.

The short version - yeah you need to contact your ISP and ask them to change the record

There is no delegation of neither 1.196.198.in-addr.arpa nor 196.198.in-addr.arpa zone on the DNS servers authoritative for 198.in-addr.arpa zone.

This is either your task to properly set it up or whoever sold/rented you the address space. Sole PTR record on your nameserver is not enough, the whole delegation chain must be also in place for this to work.