Hi, welcome to the forum! I've moved your post to the International section as it's best suited to this
Unfortunately Trainline and Trenitalia are being very unhelpful with the current situation concerning the landslide on the Lyon-Turin line. Neither are taking responsibility for helping affected customers, beyond offering them a refund - which as you say, isn't much use if you need to pay more for a ticket via an alternative use.
Legally speaking you were entitled to require Trenitalia to rebook you via the quickest alternative route, at no extra cost; this is under the EU Rail Passenger Rights and Obligations Regulation (PRO). However, they are effectively ignoring their obligations in this respect and so you have little choice but to take action against them to recover your costs (when they inevitably tell you they're not paying up).
For domestic Italian journeys,
Trenitalia has an arrangement with an Alternative Dispute Resolution provider, but it's unclear what arrangements they have for international journeys. You might want to ask them about this. Otherwise you might be limited to taking legal action against them, which is understandably not an attractive proposition.
In future I'd recommend booking journeys such as these directly with the operator if possible. If you do this using a credit card and the cost of each person's ticket is at least £100, you benefit from
section 75 protection, allowing you to pursue your credit card lender (including through the Financial Ombudsman if needed) instead.
For future journeys you could also consider using an Interrail pass, which gives you much greater flexibility. This isn't valid on Trenitalia's Paris service, but it is on SNCF's TGV "inOui" service - and given the flexibility it would have set you back little (if anything) extra to use on an alternative route via Switzerland. If you were travelling from the UK on Eurostar it may even have saved you money overall.
Sorry to be the bearer of bad news!
