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What many folks find out are that "published" (web sites, brochures, etc.) UVW's aren't reflective of actual factory ship UVW's (noted on yellow sticker).
As far as the Eagle 314BHDS I'm not surprised that you are exceeding a 1,200lb loaded tongue weight. The 314BHDS is a 37ft long TT, so your loaded tongue weight should be 13% to 15% of the TT's loaded weight for enhanced TV handling of the TT. RV and WDH manufactures recommend a 10% to 15% range.
I don't understand how Jayco Service can make the statement, "the tongue weight is 1100 lbs regardless of what the printed and website"...., how do they know one's loading habits?
Example modest loaded 314BHDS: (7,580lbs UVW) + (500lbs ship weight delta) + (100lbs dealer, battery, etc) + (500lbs crgo, no fluids) =8,680lbs TT weight. So 8,680lbs at 13% to 15% = 1,128lb to 1,302lbs loaded tongue weight.
Worst case "loaded" at 314BHDS GVWR: 9,975lbs at 13% to 15% = 1,297lbs to 1,496lbs loaded tone web range.
I agree, a 1,500lb rated WDH would be recommended.
2016 GM 2500HD 6.0L/4.10
2005 Jayco Eagle 278FBS
Reese HP Dual Cam
Putnam Class V