I spoke again and also wrote to the Exportability Claim and they confirmed without any doubt that Contribution Based ESA Support Group IS exportable to an EU country.

What is an exportable benefit?

An 'exportable benefit' is a Department of Work and Pensions (DWP) benefit which you can claim while living outside of the UK. If you receive one of the following exportable benefits, you may be entitled to healthcare paid for by the UK by applying for an S1 form: State Pension.

Is ESA an income based benefit?

Unlike means-tested benefits, there is no income and savings test for contributory ESA. ... Even if you claim contributory ESA you may also be entitled to means-tested benefits to 'top-up' the amount you get.