Try a preconditioner---I prefer a one pound cut of shellac. You can get shellac in several colors from clear(blond) to dark amber. A one pound cut is made by disolving one pound of shellac flakes(available from many online dealers or in some paint/woodworking stores) in one gallon of denatured alcohol. That really cuts down on the tendency of cherry to blotch when stained/finished. Then use a three pound cut(normal strength--three pounds of flakes in one gallon alcohol) as the next coat. You can even control the color of the amber tint from the shellac by using light or darker flakes.

Dewaxing the shellac(letting the mixture set until the wax settles) will allow you to put a final coat of water based poly---for the protection factor.

The only odor will come from the alcohol, and shellac dries and cures quite rapidly---often in as little as three or four hours.

Rather than dewaxing shellac, I prefer to just buy it that way. The most economical way is to buy Seal Coat. At $22 / gal it beats buying two pounds of shellac (at $6-10 lb) and a gallon of alcohol at $15. I use it regularly under w/b finishes and it brings out the natural luster of woods like cherry. I used this on a natural (unstained) cherry set of conference tables in a law office and top coated with a w/b acrylic.

Another option from a friend of mine, with slightly different emphasis:

You can achieve the color of the cherry very easily without the use of either stain or dye by using the schedule below. Over time the natural inclination of the cherry to darken with age will only make it better.

Begin by carefully sanding thru 220g. Make sure that you remove all sanding scratches left by the proceeding grit and don't skip grits --120g, 150g, 180g, and finally 220g. Remove all sanding dust with a shop towel dampened with mineral spirits and carefully examine the MS wet wood for scratches you have missed and for evidence of glue squeeze-out that must be removed.

Next, apply a liberal coat of boiled linseed oil (BLO). I'm talking here about applying a thick, syrup on pancakes application. Allow the oil to soak into the wood for about 25-30 minutes and then wipe away the excess and buff the surface as "dry" as you can with a clean towel. Then, set the piece aside for a few days to allow the oil to cure. The towel is a spontaneous combustion hazard so do not wad it up and throw it in a corner, but hang it up to dry.

After a few days apply a 2# cut of garnet shellac mixed fresh from flakes. Simply brush it on with an inexpensive natural bristle brush with as little over-lap at the wet edge as possible and by keeping back-brushing to a minimum. Keep an alcohol dampened soft cloth in your free hand to pick-up runs and sags as they occur. The freshly mixed shellac will dry very quickly (25-30 minutes tops).

You can then lightly sand, just enough to smooth the surface, and apply your varnish topcoat. For a light-colored varnish that will not darken itself over time use a soya oil based alkyd resin varnish such as McCloskey Heirloom. For a somewhat darker look switch to a tung oil or linseed oil based varnish. I prefer Waterlox Original, a tung Âoil-based phenolic resin varnish.

Here is a link that might be useful: another variation of finishing cherry by same author

If you are going to use garnet shellac anyway, mix some extra and cut it from 2# to make the wash seal coat.

The Class I Biological Safety Cabinet provides personnel and environmental protection, but no product protection. It is similar in air movement to a chemical fume but usually has a limited fixed work access opening and the exhaust air must be HEPA filtered, to protect the environment. The Class I Biological Safety Cabinet is designed for general microbial research with low-and moderate-risk agents. In addition, the work zone, which type is much smaller than a chemical fume hood, is geared to working with biological whose process/producers dictates the use of moderate amounts of volatile liquids, that are suitable for recirculating cabinets

Here is a link that might be useful: wood

FYI, shellac is not oil based, and is nontoxic when dry. It dries rapidly through the evaporation of its solvent - alcohol.
Shellac can be tinted with dyes and other colorants. Since the dry time is very short, exposure to the VOC would be minimal with no long-term offgassing. Oil finishes like Watco or Waterlox offgas for an extended period.
Casey

Thanks for the suggestions. I am planning to stop by the Woodsmith store later this week to look into the shellac. Can that be used under the danish oil as a conditioner to prevent the blotching or would the shellac alone be sufficient? From what I gather from above I can use a water based poly over either the shellac or the linseed oil??

I am trying to keep this project simple, fast, and non toxic yet hope to end up with beautiful, durable doors that somewhat hold up to the abuse of three young kids! Hopefully I'm not asking much.

The shellac must be dewaxed if you are going to top coat with a polyurethane or water-borne finish. To be safe, I'd use the shellac as a barrier / binder coat between oil-based finishes and w/b finishes. You stand a chance of poor adhesion, other wise. If you are mixing modes, you also need to make sure lower layers are completely cured (several to three weeks) before switching.

Danish Oil can be anything manufacturer wants to put in a can and call Danish Oil. Usually it's a mixture of thinner, boiled linseed oil, and varnish. One common brand is (more or less)
65% thinner
23% BLO
12% Varnish and other additives (coloring, driers, etc.) The linseed oil component will not serve much function if applied over shellac, as it has no where to soak in.

Run trials on scrap. Test, test, test.