Hello "Finnair" experts!

I've booked a Finnair return flight through mytrip.com (TLL-HEL-BKK - BKK-HKG-HEL-TLL), one air ticket is a whole my family 6 people. 3 adults, 2 children and one infant.
The names are just switched between first and last name (wrong order). For all six passengers. Spelling is exactly the same as shown on passport.

I have called twice and had very little help from Finnair call staff.

I have also called the travel agency (mytrip.com) who told us, that the name correction procedure is quite difficult and can be proceeded with issue new tickets for the same flights as you have originally booked. The price for now on for new tickets about 4000 EUR and can be changed depending from availability. The penalty 25 EUR per passenger will be apply for authorization name correction from airline side, except infant. Service fee of our agency 19.99 EUR per passenger, except infant.

Summary I need to pay price for new tickets plus 125 eur penalty for the name correction + 99.95 eur service fee of agency.

I have sacrificed and saved to be able to afford this trip and my accommodation is also non-refundable. These non-refundable options of course, are the cheapest deals. I understand fare rules for various levels of service are to prevent transferring tickets to other people, however, this is not my situation.
The only solution they could offer was to email to see if the staff at the airport would let us through, however, even if they say they will let us through there is no guarantee that we will be allowed to board on the day when we arrive.
I am so very disappointed and can't help worrying about my situation. The call staff's recommendation was to cancel and re-book a ticket if i want to be assured of being able to fly. I am not a rich holiday maker. I simply do not have the money to afford to buy another set of tickets.
If I can't board, my holiday is ruined and I will have lost almost ten thousand euros. Clearly the fare rules shouldn't apply to this situation.

I am very stressed and upset about this and hope to find a better solution than just "hoping" I can board the plane with my family. I don't know what else to do at this point.

I totally understand that the mistake is mine on the incorrect name/surname reversed - no excuses. I am more than willing to pay small penalties.

I'm hoping that someone from Finnair airlines would notice this issue and resolve it for me without a cancellation

Right now, i have 3 weeks till my travel, i am wondering if it is fine to travel without changing it?

please share your experience. i really need your help and any help is appreciated!!!