Can the bronchodilator exaggerate the tachycardia?

Yes: This is a known side effect of this class of medication.

Many effects: Remember that many bronchodilating medications work by stimulating certain receptors in the lung. But, these receptors are located elsewhere to, and have different effects when located in different organ systems. So while these meds may dilate the smooth muscle of the lungs, they can lead to increased heart rate when acting on the receptors around the heart.