Just a couple idea:
1. Use the hotspot feature of your cell phone. You don't need anyone's permission, yet you need to make sure that your phone plan supports it, and doesn't add extra expenses (that you are not willing to pay, if there is one). Other things to consider is if it draws from a data pool the spouse may not like that fact that now they can't stream their favorite show over their phone.
2. Speak with your bishop/branch president, and see if they will grant you access to the leadership/hidden network. This is another WiFi that does not broadcast it's SSID, you need to know about it, along with the password, that is forced to change every 90 days. This is mainly reserved for when your unit leadership is meeting, or on the fifth Sunday and they are sharing a presentation, or even if they are sharing a church broadcast (think fireside or general conference).
3. Speak with your unit leader (bishop or branch president), along with the STS, and see if they can make a case to the FM on your behalf to have an Ethernet jack installed in your classroom, along with a unmanaged, yet mounted, five port switch. Things connected physically are given permission as they are in the building, where as wireless access could be gained from outside in the parking lot. This is a case where if you have physical access it is assumed you have permission to be in the building to begin with, and not just drive-by wifi hacking. If the switch is not mounted then it needs to be secured when not in use, meaning it would be kept either in the media center or a unit leader's office, and you would need to check it out every time you need to use it.
3a. If you already have a room wired to the LAN, and hence the internet, then it may be that you only need a switch to share the connection.
3b. You would have to figure out how to connect the tablet and phone to the switch, yet that should be pretty straight forward nowadays, the laptop is easy as it either has an RJ-45 plug built in or you get an USB to Ethernet adapter.
I think #2 is your best option. Know that your unit leader, and the STS, would most likely want to go over some guidelines or set restrictions, with you. The first being that you are not allowed to share the password with anyone. Next would most likely be that you could only use the hidden network while conducting your class, and agree to use the public network the rest of the time (try telling that to your phone or laptop - good luck). Even if they don't mention it, maybe you should say this is what you are willing to do, and it is to help the youth of the church.
Thank you for taking your calling seriously, and I hope you manage to get things worked out.