One way to do this would be to work out exactly what % equity you have now i.e agree an actual market value now with your partner(get a proper valuation done now that both of you agree), work out what the actual amount of the mortgage is as a % of that value and the balance is your equity. E.g based very roughly on the figures in your first post the mortgage is circa 77% and your equity right now 23%.
Then you agree to go 50/50 on the mortgage so you get half of the 77% and your partner the other half.In other words your partner owns 38.5% and you own 38.5% plus 23% =61.5%.
You then have your solicitor draw up a co-ownership agrrement that covers this, and also these %'s go into the deed of transfer into joint names.
You should thereafter ensure that apart from the mortgage which you pay 50/50 all other major items of expenditure are split 61.5%/38.5%(you paying the greater amount.
This way your existing equity is safeguarded.
Alternatively have your partner pay you now 50% of the equity now (circa €37,500) and then you can own the property 50/50
These are merely suggestions and ultimately you have to decide what works for you. That said its a bit surprising IMHO that your solicitor didnt suggest a few options.
For both you and your partners sake whatever you decide you should have a co ownership agreement drawn up. In the unfortunate event that you split up in the future then you avoid a lot of aggro (and increased ill feeling)if a good agreement is done while you both could still agree. The agreement should also deal with what would happen if one of you were to pass away(in addition you should make wills), whether in splitting up one or other of you has an option to purchase the others interest as opposed to the property having to be sold etc..