ESA basic rate during appeal- is this a public fund?

During an appeal for Limited Capability of Work is the basic rate of ESA income-based ESA and therefore a public fund? Client is only entitled to C-B ESA and so does this mean he should not be claiming the basic rate during an appeal as this may jeopardise his right to stay in UK?

As far as I know, the ESA paid pending appeal stays ESA (CB) and does not turn into ESA(IR). I had some clients who would not have been entitled to ESA(IR) and continued to get ESA pending their appeal.