Cut bacon into 3/4-inch-thick matchsticks and cook in a heavy skillet over moderately low heat, stirring, until crisp but still chewy, 6 to 8 minutes.
Transfer with a slotted spoon to paper towels to drain.
Pour off fat from skillet and add 1 1/2 tablespoons oil, mushrooms, and 1/2 teaspoon salt. Cook over moderate heat, stirring occasionally, until mushrooms are golden, about 8 minutes, then cool.
Whisk together lemon juice, vinegar, mustard, remaining 1/2 teaspoon salt, pepper, and remaining 2 tablespoons oil in a large salad bowl until blended.
Add pea shoots and spinach to dressing and toss to coat.
Add bacon, mushrooms, chives, and radishes and toss again.