This is not intended to be a 'smart' answer: you could lower the garden path. Dig down instead of building up around your trees.

For the perennials and bulbs - dig them out as if you were going to thin and renew, and put them back into the new bed.

Also, if you can reduce the amount of excess moisture entering the garden area. Duct it in some way.

(Proposals to nudge the thinking along...)

One has to be careful around established trees, since most do not do well if you suddenly add 3" of soil on top of their root system. However, 3" - 6" of coarse organic mulch will gradually decompose - raising the general level of the bed, while not upsetting the tree (keep the mulch at least 6" away from the trunk). By adding more mulch every year, you could have the height raised about an inch per year. Vertivert's right about lifting the bulbs, although most common bulbs will adapt to a gradual depth change, just being slightly later to bloom, and not multiplying as rapidly.

Another thought is to deeply edge the bed [like a miniature drainage ditch], although you do want to arrange the mini-ditch so the water will run off, away from the bed. A local friend took care of his problem by making a pebble-filled dry stream wander through the wet area. I don't remember the details other than that he dug a trench and filled it with very coarse rocks; topping with white gravel and local stones; it wanders more or less diagonally across a wet area and ends at the edge of a stream. He planted bulbs, hostas, and various grasses along the edges... very realistic. What I especially admire is his little Japanese-style bridge that crosses the 'stream' from the edge of the lawn into the shady, moist area. Fair to note, though, that he spent about 3 years doing all that.