If his relationship with FC is good, try to nab as much from their routine as possible, even linguistic habits or pet names for drinks and food. We found it quite endearing to say "what are you crossing for?" Rather than "what are you having a tantrum for" and he responded well.
In terms of speedy and public transitions, I am a strong advocate for making sure they have exactly the same food and drink (down to the brands), and try to take note of the types of things he is wearing.
You could do something like take a tiny treasure/item to him each time and get him to bring it back to you and tell you what it did.
We built connection by doing different games - playing peekaboo, letting him wear our shoes, playing mirroring body language games, letting him pour water from our cups to his etc. anything that meant he had to interact with us. We took him jenga and let him build with us, turn taking and stuff that could be done without words as well because he was quite quiet and refused to speak to us much, but he enjoyed that. He also enjoyed playing hide and seek but if it is all in public that might be a bit much. Try to get on his level, copy him and what he is doing, you could make it into a game or just allow him to come a bit closer each time if he is shy.
Could you do something to make him laugh, like wear a t-shirt with a funny character on it that he likes, or matching shoes with him, do a 'magic trick' of bringing a coin out of his ear? Ours love that sort of nonsense.
We got a talking book with photos in and pictures of their bedroom with the transition toys and played a very simple 'wheres wally' type game with them each time we saw them, and then they took it back and they could press the buttons and play us saying the same thing. You can Amazon prime them for expediency.
Smell is important too- can you give them a jumper of yours that you have slept in for them to sleep with or cuddle until they come to you?
I feel like I'm a walking advert for them but you can get a Tonie box that you can record your voice onto a little figurine, so they can play you reading them a story and get used to your voice. It's also easy to charge and doesn't require adults to put it on or do it, they literally just put a small figure on top of a box, so it might mean he is able to do it more if FC aren't playing ball with showing him stuff.
Make sure you take as many photos of them and you in the FC placement or with the FC if at all possible. Even if it isnt working out for whatever reason, it's about the build being able to connect you to that place and have a coherent narrative. Could they be persuaded to record some videos that could be played to him at a later date?
In terms of settling in, keep it very much the same every day until you die of boredom. Honestly. For our eldest, we literally did the same thing over and again and it was so helpful to establish our routine and get us to understand what was a typical reaction to stuff. After 3 weeks we occasionally changed stuff up by going to a different park or adding in a visit to the corner shop. Heady delights indeed! But it meant we could really settle in. We even did the same food for him every night when he started refusing to eat. We cosseted and babied him and got him whatever nice stuff to eat he wanted, smoothies, biscuits, stuff like that. Our therapist suggested making sure he always sat with us when he ate to associate us with the nice stuff. We sat up with him at night and sang him endless lullaby songs. We played baby games like him stepping on our toes and 'dancing' and swinging him up like a baby. He slept with the lights on as well.
Do plenty of screen free activities, but also remember that TV might have been a comfort and a familiar thing for them. Cbeebies is a bit of a staple in many houses, and they might find the option to turn their brain off and watch moon and me or hey duggee comforting. Once we got rid of the idea that we wanted them to be screen free, it allowed us to build TV into the routine and it was like magic. He wanted to watch cocomelon on repeat and it was something that he found so calming that it reduced his outbursts.
Also, on a side note, If he is a crier, I can also highly recommend loop earplugs. It sounds awful, but I suffered so badly from overstimulation in the early days, I was so stressed and overwhelmed and the noise was just... Unbearable. It made me so much more empathetic when it wasn't like nails down a chalk board squealing and screeching.
Think carefully about his stuff. Has there been a plan? We had to sneak out LOs stuff out in bin bags disguised as rubbish because it was too distressing for him to see it come to ours. But that does have a knock on- he is now always worried that his stuff will go missing now. Could he pack some of his chosen precious stuff in a trunki and give it specially to you as a symbol of him making that transfer? You can pick up trunkis on FB marketplaceplace for about a tenner, or new from Halfords for £25 that I bet he would love. You could get one of something he would like. You can tow him on it around the park or shopping centre, and kids love that, give it to him to take home and the collect it the next day, getting him to show you what he chose to put in it. (As a guide, when they got to take what they wanted to grandmas for our first sleep over, he took a necklace, 3 pairs of undies, a mini monster truck and a bit of plastic tubing) but the point is you are showing him that you think what he thinks and feels is important.
On a final note, take care of yourself too. It's a difficult time, so be kind to yourself and make sure you are getting as much rest as possible.