In reply to joseph_dyball:
I tried the Lion Ridge at the start of this month, so I will probably provide the latest advice - essay incoming!
We were based in Zermatt and took the lifts across which was certainly the easiest (and most expensive??) way to get to Italy. The Swiss have just completed construction on the final gap in the crossing (opened 1st July) so you no longer have to walk down the pistes. We got Swiss half fare passes (CHF120 for 1 month) which are crazy expensive, but pay for themselves if you take 2 lifts or more. This made the lift price for a single to Plan Maison in Italy CHF84. Bare in mind that if you don't do the full Matterhorn traverse, you've got to pay this again to get back - We learnt the hard way!
We left Zermatt at 2pm, PM at 3.30pm and were at the Carrel by 8pm. We soloed the AD- approach which should not be underestimated as it is pokey in places which is why we were quicker than guidebook time. In hindsight we should probably have roped up for 2 short sections.
As has been mentioned, the Carrel is closed for renovations this summer, however it had been left open prior to the workmen arriving (relief!). I expect this to have changed, or to change imminently, so expect the Carrel hut to be CLOSED. Options for bivying by the hut are extremely limited, though this is what we had planned to do. As you would expect, this side of the mountain is much quieter than normal, though our worry that the fixed gear would be in poor condition as a result was thankfully not true.
As of the start of the month, there was A LOT of snow on the route (plus bad weather) which made the going slow. There is an approximately 200-300m knife edge section of the ridge past pic tyndall which is steep and effectively unprotected due to snow. The approach shoes conditions described in the logs sound much more appealing!
There is less fixed gear than I was expecting and it is very challenging to route find off fixed gear alone in poor vis, so pick favourable conditions. I can't speak for others, but it took us longer to descend than ascend, so set a turn round time and stick to it. Also the crux is at the top!
Seems entirely reasonable to do the full traverse in good conditions like we had planned. We left food and water at the Schwarzsee lift above Zermatt with the intention to bivi for a second night, before taking the lift down on the third morning.
Best of luck, hope that helps, and let me know if you have any other questions!
Post edited at 13:57