Should I get a UK visitor visa anyway?

You really need to review the UK reviews on remote working. The UK does not generally like it, and the status of the law is a mess.

• See Immigration Rules V 4.23 (that's vee, not five) and appendix 3

V 4.23 Visit visas, leave to enter or an extension of stay as a visitor will be subject to the following conditions:
(a) no recourse to public funds; and
(b) no study, except as permitted by paragraph 25 of Appendix 3; and
(c) no work (which does not prohibit the permitted activities in Appendix 3, 4 or 5 as set out in V1.5).

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VISITORS APPENDIX 3. PERMITTED ACTIVITIES FOR ALL VISITORS

Business – general activities
5 A visitor may:
(a) attend meetings, conferences, seminars, interviews;
(b) give a one-off or short series of talks and speeches provided these are not organised as commercial events and will not make a profit for the organiser;
(c) negotiate and sign deals and contracts;
(d) attend trade fairs, for promotional work only, provided the visitor is not directly selling;
(e) carry out site visits and inspections;
(f) gather information for their employment overseas;
(g) be briefed on the requirements of a UK based customer, provided any work for the customer is done outside of the UK.

Business – corporate
Intra-corporate activities
6 An employee of an overseas based company may:
(a) advise and consult;
(b) trouble-shoot;
(c) provide training;
(d) share skills and knowledge; on a specific internal project with UK employees of the same corporate group, provided no work is carried out directly with clients.

• See Home Office's Visit Guidance (for immigration officers):

For visa applications you should consider whether the information provided about the applicant’s purpose of visit/ activities to be undertaken would amount to prohibited work under V4.5. Permitted activities under appendix 3 must not amount to employment or filling a role even on a temporary basis. Where you consider that the information provided means that this is the case, you must refuse the application and notify the sponsor management team in case the organisation holds a sponsor licence.

At the border, you will need to ask the applicant to explain what they are coming to do and for how long. Where you consider that the information provided means that the activities amount to prohibited work under V4.5, you should refuse entry.
The reasons for refusing a visa, leave to enter or cancelling or curtailing leave must be clear and supported by evidence.

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General business activities
See: appendix 3 paragraph 5 of appendix V: visitor rules.
You should assess whether the period of leave requested is credible in view of the activities they are seeking to do, and be satisfied that the visitor is not coming to work or make the UK their main place of work. For visa applications this evidence might be included in a letter from an employer.

Intra-corporate activities
General business activities
See: appendix 3 paragraph 5 of appendix V: visitor rules.
You should assess whether the period of leave requested is credible in view of the activities they are seeking to do, and be satisfied that the visitor is not coming to work or make the UK their main place of work. For visa applications this evidence might be included in a letter from an employer.

I know you say "incidental" but they don't think that means what you think that means. To them "incidental" is an hour a week. You want to work 40-ish hours a week, that's not incidental.

These are the latest (Jan 2020) rules, and it's not looking good for our remote worker. The instructions to IO's seems to be to turn you away if in doubt. That is, the instruction seems to be to figure out what you'll be doing for 3 months, and if it sounds like work, to bounce you. It would not be sufficient to have no good explanation for what you intend to do for 3 months; if you don't have a full docket of allowed activities, they will suspect. Especially when they ask about your employment; do they really believe you are on a 3-month sabbatical?

It will also hurt you if your luggage is well-kitted-out for remote work, because when you look like a bit of a wanderer (i.e. have weak home-ties), immigration is sensitive to seeing "tools of trade" in your baggage. One woman got turned away for having 10 copies of her resume!

It also cannot be overstated the importance of leaving the country timely and not overstaying. An overstay will disrupt future visits.

I'm also concerned about tax issues, since you are earning the money while your feet are in the UK, but my brief research says you'll be fine if you DO NOT overstay 90 days.

Oh, and if you apply for a visa and are refused, that's your answer.

Your idea is to come anyway, and hope for the best. One of the dumb things people do is assume immigration is dumb. In this case you think the border guards have no memory, and you can strut up to a random border guard and get a fresh start. Nope. When they scan your passport, all your data will pop up. They'll ask if you saw the letter saying you were refused, and want to know why you came anyway. And you'll have to pay for a flight straight home.