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My oldest child turned 4 in June. According to age, that would place my 4 year old in LHTH, but he is way past needing to learn letters and sounds (he's starting to learn to read simple stories), he can write letters/numbers (although not always placed correctly on the lines), he can do some simple adding and subtracting, and he can focus for quite a while on stories/activities that interest him (For example, he can sit and listen to whole chapters in chapter books or spend 30-45 minutes putting a puzzle together.) In looking at the placement guide on the HOD website, these abilities would place him in LHFHG, but the age range for that book is 5-7, so I'm not sure which guide would be best for him. Any thoughts?

It does sound like your child could handle LHFHG. You could do LHTH and add on handwriting work, math from LHFHG, and reading. This would give your child time to grow into the maturity needed for more school work. The history in LHFHG is a high level and the storytime, without many pictures and longer readings, is really more appropriate for a 5-6YO. LHTH is a great Bible literacy year. You read and study through the whole Bible (I would suggest reading from scripture rather than using one of the children's Bible options).

If you felt LHFHG was really a much better fit, then I definitely highly recommend taking it very slow. 3-4 days a week at the most. Use the other days for nature walks, field trips, "home ec" (cooking, cleaning, learning life skills). There are so many great things to do that don't involve books.

Or you could just keep on doing what you are doing, letting him learn naturally and don't do formal guide. There are plenty of years for that. These are the years for lots of play time!