Is there any computer program to calculate the dose of the whole decay chain to get a picture of the artificial radiation and supports logging. I don't want to look up all the individual numbers and calculate it manually.
Yes, there is. MCNP will do dose calculations, among many many other things. It is a stochastic code, meaning it does random flights and interactions of radiation, which would be beta particles in your case. Those are a good bit more tricky to model than gamma rays because they bounce around a lot more and also have continuous interactions as a charged particle.
Radioactive decay is also handled with the MCNP code, but not directly. The national labs use the Monteburns code for this purpose and it is linked with ORIGEN2, although I should specify these can change depending on the version number. You would likely get MCNP5 or MCNP6 if you ordered it now.
In order to use these codes, you may submit a request to RSICC, although ideally you want to have a US university affiliation, you can also obtain it working in industry although probably at a higher price. You also would really need someone on your team with a few years on experience working with these codes.
The cost of accurately obtaining the dose rate numbers will certainly affect that approach you take, as the acquisition of data, even from models, has a very real cost associated with it. It's likely that your lab will not have the budget and you will instead consult a textbook and try to combine rough estimates for the various sources you have.
There are a variety of tools available for managing radiological dose to workers in a lab. In the US we would expect that a lab working with any significant sources would have an ALARA policy, which dictates minimizing exposure through the fundamentals of time, distance, and shielding. In a nutshell, ALARA reflects the precautionary assumption that there is no "safe" dose so any unnecessary or frivolous dose is unacceptable. Modeling, direct radiation detection, and TLD devices are common tools for radiation safety, but these are not necessary for all labs and any combination of these measures may be employed based on the needs of the specific lab. Many sources are low enough hazard that none of these will be employed. However, an understanding of radiation and the types of sources you're working with should come before any of these options. If you are the primary person in your lab responsible for a source and you do not know the hazard level or the appropriate precautions that should be taken for it, then that would be a serious problem and you should consult the management in your organization or the nuclear regulating body in the area in which you reside.