They provided some rules that I could pass to my new host that they could add to the module to let Perfect Dashboard communicate with the sites correctly.
It took about 48 hours after the rules were applied for Perfect Dashboard to start functioning correctly. Not sure why but at least everything is again as it should be.
If you're interested the rules to add to mod_security are as follows:
SecRule REMOTE_ADDR "^46\.101\.7\.25$" phase:1,nolog,allow,ctl:ruleEngine=Off
SecRule REMOTE_ADDR "^188\.166\.167\.199$" phase:1,nolog,allow,ctl:ruleEngine=Off
SecRule REMOTE_ADDR "^46\.101\.105\.144$" phase:1,nolog,allow,ctl:ruleEngine=Off
As an alternative it was also suggested the code below could be added to the root .htaccess file:
SetEnvIfNoCase Remote_Addr ^188.8.131.52$ MODSEC_ENABLE=Off
SetEnvIfNoCase Remote_Addr ^184.108.40.206$ MODSEC_ENABLE=Off
SetEnvIfNoCase Remote_Addr ^220.127.116.11$ MODSEC_ENABLE=Off
If you have a similar problem then I hope one of these two methods can provide you with a solution.