Your UK accountant is wrong, or you misunderstood him.
1) Domiciled and Resident people must pay tax on income whether remitted or not.
2) NonDomiciled and Resident people pay tax only on remitted income.
3) NonResident people don't pay tax on remitted income, whether Domiciled or not.
In colloquial terms, Domicile is where your heart is, Residence is where the rest of you is. So, since you're from the USA, your heart is back there, but the rest of you is here.
4) A huge gift from your father will trigger US gift tax obligations. He should not make the gift without speaking with his tax advisor.
5) You selling stock and then giving a gift of the proceeds to your father would trigger a US gift tax obligation too. You shouldn't do that without considering the long-term consequences of such an action.
6) There are ways to remit sums from capital gains into the UK without triggering remittance tax. This requires a planning session with a qualified UK tax accountant who handles remittance taxation. You supply him or her with a list of ALL your assets held outside the US, listing each stock individually. For each item on the list, you provide the purchase date, the purchase price, and the current price. S/he then computes how much taxable capital gains would result from the sale of each item in the UK, including
currency exchange differentials,
UK tax reliefs (special tax breaks for capital gains), and
credits for US tax paid on the gains (assume 15% on everything owned >1 year).
S/he then ringfences the items that will generate the lowest remittance tax, which could very easily amount to zero.
That's why folks pay accountants - not to run numbers after the fact, but to run them beforehand so they owe no tax.