The easiest approach is to drill a smaller hole first, say 1/8" and then go to the size required. Wrap masking tape around the drill bit to mark the correct depth.. Oak has a grain that is "open pored" thus you hit high and low density sections that want to make the bit wander.. less prone to this if you drill a smaller pilot hole first.
but aside from that you may have an issue with strength. I have used a similar type of connection with handrail installations and by the time you drill the correct size of hole for the lag, you typically have very little wood left to play with on either side.. if you are able, I'd take some of the leftover pieces of 1x3 and glue them on the ends to reinforce or go to a thicker material .. say 5/4.. or a full 1" in thickness.. will be alittle harder to find and may have to visit a local woodworking shop .. good luck Karen
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