That’s quite the plan there!
I’ll fill this post in bit by bit.
All of these assume each section is the only travel on each day, if you combine some then chances are a pass day will be the better option.
Luton - Bath
Eurail Pass - Luton - London St Pancras and London Paddington - Bath, underground or bus between the 2 London stations not included, use contactless bank card for London travel
Bath - Swansea
Pass. You could save a small amount by buying fixed journey ticket online in advance.
Swansea - Tenby
Regular ticket, buy on the day for £19, advance booked for a fixed train for less.
Tenby - Portmeirion
That’s going to be a long journey no matter what you do, Links between South and North Wales are slow.
Quickest will be by train Tenby - Carmarthen, bus Carmarthen - Aberystwyth, then train or bus Aberystwyth - Minffordd which is the nearest station and bus stop, about a mile from Portmerion. That will be 7-8 hrs
By train only you would be going via Cardiff and Shrewsbury, taking 9hrs or more.
By train only a pass day.
The bus is not included in Eurail so the cheapest for that option is likely to be a regular rail ticket at £35.70 which includes the train - bus - train connection.
Portmeirion - Traeth Porth Wen Brickworks
There is a day bus ticket for North Wales for £6 that will cover that.
https://traws.cymru/t1c/use-1bws-ticket-north-wales
Traeth Porth Wen Brickworks - Bethesda
As above
Bethesda - Liverpool
Bus to Bangor, Train changing in Chester to Liverpool.
Train is £32 regular ticket,or cheaper fixed train advance or use pass day.
Liverpool - Manchester
Buy regular ticket, less than £20
Manchester - York
Regular ticket £35 or use pass.
York - Edinburgh
Pass day.
Edinburgh - The Kelpies
Regular ticket, long walk or bus from Falkirk train station.
The Kelpies - Inverness
Pass day
Inverness - Glenfinnan
No remotely direct train for this, Bus Inverness - Fort William and bus or train Fort William - Glenfinnan would be the most direct however the rail lines Inverness - Kyle of Lochalsh and Mallaig - Glenfinnan - Fort William are 2 of the most scenic lines in Britain, it is well worth spending extra time to do these. It is possible to connect from Kyle to Mallaig by bus and ferry. If you are interested in this give a date and I’ll check the bus and ferry times, they seem to change constantly.
Glenfinnan - Glasgow
Pass day. Another very scenic line.
Glasgow - Belfast
3 options for this.
Sail-rail - train Glasgow - Ayr, Bus Ayr-Cairnryan Ferry Carnryan-Belfast Port (and bus to Belfast Centre which is not included in fare)
£35 regular fare
Pass is no good for this
https://www.stenaline.co.uk/rail-and-sail/to-ireland
Bus with Citylink, Bus to to Ferry terminal and second bus from Belfast port to city centre
£32
https://www.citylink.co.uk/
Bus with Hannon Coach. A bit more at £43 but with this service the bus drives onto the ferry so you can load your bags and not have to unload them and carry them on the ferry. This is the best option IMO.
https://hannoncoach.com/belfast-to-glasgow-with-hannon-coach/
Belfast - Donegal
Train or bus but train is more scenic Belfast - Derry
No point in using a pass day for this as regular fare will cost less.
Derry - Donegal by bus, there are no trains in this part of Ireland.
If you mean Donegal Town then it is a direct bus, if it is somewhere else in County Donegal then it may require more planning.
https://www.expressway.ie/
Donegal - Galway
By bus, as above
https://www.expressway.ie/
Galway - Killarney
Use pass on train with multiple changes https://www.irishrail.ie/en-ie/
Or a bit quicker and just 1 change by bus, Expressway again as above,
Killarney - Cork
Similar options, Bus mostly quicker than train, if using train €15 fare available on Irish Rail website.
Cork - Dublin
Train, use pass day or buy from Irish Rail website, similar cost.
Nice - Monaco/Antibes/Cannes/Grasse/Villenfrance Sur Mer/etc
Nice - Marseille
Marseille - Arles
Arles - Lyon
Lyon - Paris
Paris - London