Hi, I spoke to Conga support and was able to use
{IF "{MERGEFIELD _RECORDTYPE}" = "Third Party Unit" "{MERGEFIELD IMAGE:Bedroom_Generic_URL:W=250}" "{MERGEFIELD IMAGE:Slideshow1_URL:W=250}"

I also had the wrong label. I was using the API name Apartment_Record_Type__c when I needed to use the Conga label of _RecordType. Thanks for your help.