Make very accurate solutions without weighing
Are there any other ideas how to get exact concentrations using small amounts of the desired substance?

Every mass spectrometry lab should have microbalances to prepare samples and standards. A lab which can afford an LC-MS can also afford a semimicro balance or a microbalance. Such balances are meant to weigh 0.01 mg to 0.001 mg (with extreme caution).
Two simple options:
(i) Since you are apparently not the first user of LC-MS system in your group, ask how other senior researchers prepared standards in your lab?
(ii) Another (crude) option is that you try weighing by difference. You will tare your analytical balance to zero. By an analytical balance I mean a balance which reads to four decimal places in grams, this is the most commonly used in research labs. Place your standard container on the balance and read the mass. Carefully transfer approximate amounts to your desired container in which the standard will be prepared, and note the weight of the standard container again. The change in mass corresponds to the amount transferred. Practice it with some commonly available substance to see if this works and if your balance is sensitive enough or not. Balances age with time.