One of those "aircraft bits" should work fine for you. Just because the bit is 6" long doesn't mean you have to actually "use" all six inches. You should be able to find quality bits from multiple sources on the internet. (Maybe not Amazon...) You might find some locally at professional-level tool vendors if not at the BigBox stores. (A little help)
I wouldn't expect to break any bits, but I suppose it could happen. Everybody has bad days. Since you may not be able to jump in the car and pick up another one, you might order an extra. (Maybe if you don't use it you could send it back.)
Aircraft extension bits get their name from their use primarily in the aircraft industry in the years coming up to WWII.
Aircraft drill bits are long drill bits originally designed for use in the aircraft industry where they often need to drill holes reaching through deep cavities, and then through thin materials. Here's more about them.
The pilot hole size you'll want will be something like 9/64" or 3∕16" depending on the hardness and moisture content of the wood you're drilling into. Both of these factors can affect the fit of the screws in the pilot holes. Here's a write-up that goes into more detail about it.
I'd do a test or two starting out with the smaller diameter bit and make the hole larger if it proves to be too snug to drive the screw easily. It's always easier to make the holes bigger than it is to make them smaller.
Another idea: There are self-drilling lag screws manufactured to not need pilot holes. They tend to be more expensive, but from what I understand they are roughly structurally equivalent. I've used these, Spax 5/16 x 6" on a couple of projects with satisfying results. This may be an alternative for you, maybe not.