The FEIE only covers "earned" income, like it says on the tin. For passive income, like dividends, capital gains, and interest, you'd need to use the Foreign Tax Credit, to take credit on your US taxes for taxes paid to the UK (or vice versa, if it's US income, like dividends from a US fund).
That said, if you're looking at $15k in gain, there's likely no UK tax due, you'd use the £12,300 capital gains allowance in the UK. Because there's no UK tax to pay, there's no credit to take on US taxes - you may actually owe US tax on this. That's one of the downsides of the tax treaty, we're always subject to the
higher of the two rates. If you've got income that the US taxes but the UK doesn't, you wind up paying US tax on it.
Also worth noting that the UK doesn't do FIFO, LIFO, or SpecID, there's only one method of calculating gains. Definitely read the rules before trying to calculate it, but at a very high level it's a pooled average cost.
Transferring the money is not taxable in either country. Wise may ask for additional verification of the source of the funds, as part of anti-money laundering (just a quick email describing where it came from if they ask, maybe a brokerage statement). And if you haven't been filing an FBAR, you'll need to. But no tax.
Taxes on the house are pretty straightforward. UK has no tax on sale of primary residence. US has a generous exemption of $250k of gains ($500k if MFJ). So unless you do very well on price appreciation of the house, nothing to worry about there.
The fun one can be US tax on phantom gains on the mortgage. Very high level, the IRS sees the mortgage as a second transaction, separate from the house. If you borrow £100,000 at $1.3 to the £, you've borrowed $130,000 (the IRS only thinks in USD). If the exchange rate changes to £1.2 to the dollar and you pay back all £100,000, you've only paid back $120,000 - you've made $10,000 profit, according to the IRS. Of course, if the exchange rate goes the other way, that's just too bad for you, the IRS doesn't allow the loss. Silly, nasty part of US taxes for those abroad, a bit more detail here:
https://www.bogleheads.org/w/index.php?title=US_tax_pitfalls_for_a_US_person_living_abroad&mobileaction=toggle_view_desktop#Phantom_currency_gains