Madrid to London w/ Expired 90/180 days stay as US citizen

You seem to be trying to pretend that an overstay is no big deal.

It's a big deal. Fix it.

Someone has identified an extremely affordable flight, and you need to be on it so you are in UK soil by midnight on the 17th.

You created a belief in your mind that Spain is "relaxed" about overstays. You are missing the entire point of what an overstay is. Perhaps you think the consequences for the overstay occur at exit, and they make you pay a $500 fine or if you only have $40 just settle for that, or whatever, and the thing's forgotten. That is wishful thinking.

In reality, if you overstay, they will cheerfully let you leave Spain, no problem. (Since that is, after all, what they want you to do: Leave). The consequences will come later. When you are trying to enter another country, or revisit the EU again. The countries will observe that you have an overstay. If they interchange data the overstay will flag up on your record along with the officer's notes when you interivewed in June (if your story at that time doesn't match up with your 100ish day stay, they'll know it).

This will result in refusal at the entry. You'll have paid round trip airfare, hotel, concerts, whatever - you'll arrive, get refused, and be forced to pay full boat list price for a flight home. That happens pretty quick if you flew from the USA, but for example you'll be flying from Spain to the UK. The UK can toss you in jail until a flight to USA can be arranged.

Then if you want to fly in the future, you'd need a visa because of the refusal. Visas are expensive.

If you're a YOLO kind of guy who doesn't think about the future, no worries, it's all cool, man. But if you care about your money and travel rights, it is very, very stupid to overstay simply to score a cheaper airfare. The down-the-road costs will make it the most expensive airfare in history.

Another poster confirms last minute bargain flights exist. Be on one.

Life will be much easier down the road if you take care of this today.

Americans who arrive in the UK with no definite plans, no return travel planned, and little money are often questioned extensively at the border and may be denied entry and removed.

In your case, if you're low enough on money that you need to wait two weeks just to be able to afford a flight to the UK (*) it seems that it would be extremely difficult to convince an immigration officer that you can afford to sleep and eat in the UK for three months under the usual "no work / no recourse to public funds" conditions, and then travel to somewhere you'll be allowed to go. This is in addition to the Schengen overstay, which they may or may not care about in the circumstances.

*) which must be low indeed since Google Flights finds a connection MAD-OPO-LCY on the 17th with TAP Air Portugal for less than 100 euro.

Of course there may be additional facts that you have not described in your question or the comments which give you a way out. But as written here, it sounds like it's a definite possibility that the result of going to the UK would be a stay at an immigration detention center followed by a transatlantic flight on Her Majesty's dime (which they might attempt to collect from you afterwards).

This might work, but it strikes me as an astoundingly bad idea. The consequences of failure are too great to justify the benefit of saving a few hundred dollars in airfare. Even if you were leaving the Schengen area after 87 days there instead of 97, this plan would be risky. The chance of failure is rather higher if you overstay in Schengen, however, and the consequences of failure rather more grave.

1. There's a chance you'd be caught when you leave Spain

   Sure, Spain is known to be lax, but that doesn't mean you won't get a fine or a ban. Is the fine smaller than the money you'd pay to fly out before your 90 days expire? Are you willing to risk not being able to return to the Schengen area after 90 days elsewhere?

2. There's a good chance you'd be denied entry in the UK.

   They will probably put you in secondary inspection and scrutinize your application for admission closely. If they do this, they may uncover the Schengen overstay. A Schengen overstay by itself does not pose a barrier to entry into the UK, but it allows the immigration officers to find that you have little credibility with regard to your stated intention to abide by the restrictions of your prospective leave to enter the UK. This means they don't have to trust anything you say, which is pretty much fatal to an application for entry.

   They are also going to want to know how you're supporting yourself for at least nine months without working, since you do not have permission to work in the Schengen area nor in the UK. The UK does not permit visitors to work remotely for foreign employers while they are visited the UK. If they decide that you're planning to do this, they will not admit you.

   If they deny entry, they can't send you back to Madrid because you cannot be admitted into the Schengen area. Instead, they will detain you until they can arrange to send you to the US. Immigration detention in the UK is truly awful. You might be able to avoid it by buying yourself a ticket to the US, but that will certainly cost rather more than the money you're saving by not leaving the Schengen area on time.

Unfortunately, I cannot put numbers on the probabilities. They depend to a large degree on your "personal impact" and "articulation skills," which are unknown to us. But you should be aware that by trying to save a few hundred dollars on airfare you may in the end have to spend a few thousand dollars on airfare, especially if you end up flying back across the Atlantic after 90 days away from the Schengen area. On top of that, you risk having a truly bad day, week, or several weeks.

You ask:

Are there any forms I can fill out to maybe extend my stay by a week to avoid any hassle at the airport?

You should ask at the oficinas de extranjería. The foreign ministry has a relevant page, which links to a more specific page on residing in Spain. That page notes that you can apply for a temporary residence permit if you are self sufficient. I could not readily find a list of the conditions for issuing such a permit, but I think it requires you to have a type-D visa; you might want to ask at Expatriates.

This page advises specifically against entering another EU country after overstaying in the Schengen Area.

However, as you entry and exit stamps are not linked to a visa, you might get away with it. Entering Spain again after 90 days could prove more difficult.

See: The Truth About Schengen Information System (SIS)

Beginning in 2021, ETIAS might make overstay more difficult. Overstaying now could also affect future ETIAS approval.