Sweden has 8.4 asylum applicants per 1000 residents whereas Hungary, the second ranking European country, has only ~4. Most other European countries have 0-2 asylum applicants per 1000 residents (source).
What can Sweden do to make the distribution of asylum applicants more even across Europe? What can the European Union do?
Remember that the Dublin regulation and the Non-refoulement principle cannot be violated.
The power of the purse and law are two ways Sweden and the EU can address the issue of asylum distribution across Europe. From this standpoint, they may both act to help redistribute wealth and homogenize legal statues so that all countries become more similar and their desirability as an asylum country is more equalized (while not a preferred approach, Sweden could also make itself a less desirable country by imposing harsher treatments to asylum seekers and citizens). Second, the EU and Sweden may help set agendas that reform economics and legal standards in a asylum seekers' country of birth, particularly those that migrate to Sweden or Hungary. Additionally, the Dublin regulation could be repealed and an EU commission could centralize asylum authority. A campaign headed by Sweden could help make the latter a reality.
The rules of the Dublin system being what they are (even if it does not work in practice, more on that below), since you ruled out changing it, the answer to your question is unambiguously “nothing”.
Discussions or calculations of what a “fair burden” might be are completely beside the point. Very few people seeking asylum want to be assigned a destination and most of the countries in the EU certainly don't want to go along with a plan like that. And such a system can't be implemented if the countries that are supposed to take more asylum seekers in do not accept to relinquish their right to turn these people back to the country through which they entered (i.e. revise the Dublin system, at least for some asylum seekers).
Even the very notion that a country could refuse to examine a genuine claim to asylum because the person can be deported to a reasonably safe third-country is somewhat problematic in itself (of course, it's already what the Dublin system does and common practice in some countries but that does not make it a sound idea from a human rights or international law perspective).
To understand why it would barely make a difference, it's useful to consider the practical problems with the Dublin system. First, the whole system is based on the notion that European countries trust each other to treat asylum seekers fairly. But that's simply not the reality. The situation is so bad that Sweden stopped deporting asylum seekers back to several countries (even if it has the right to, under the Dublin system) and the EUCJ issued a decision going in the same direction, effectively shutting off the Dublin system, at least for some countries/categories of asylum seekers.
And that was in 2010 so before the mess we created in Libya and the worsening of the whole Lampedusa situation. Five years later, there is still no solution, thousands of people are dying, France is basically reestablishing controls at its border with Italy and EU countries have only begun to speak about temporary fixes.
The second problem is that some countries are completely overloaded with migrants showing up at their borders, most notably Italy. Consequently, they apparently stopped registering many of these people, undermining the Dublin system in another way. Meanwhile, Hungary just announced that they simply won't take asylum seekers back anymore, even though they have to according to the current rules. You can devise another mechanism to spread a certain number of applications and/or refugees between EU countries, but it does not help solve the issue of the many people who will inevitably be left out of the new system.
Further North, there are hundreds of people stuck in France and hoping to reach the UK. Unlike the sans-papiers of the 90s, those are not undocumented migrants who want to make a living in France and simply ask for a legal status but people who arrived recently and are for the most part hoping to go elsewhere. Because they are already on French territory and cannot be sent to another country (possibly because of the registration tricks mentioned above), many of them could lodge an application for asylum in France. But they don't want to. Here again, abstract notions of how many applications should in principle be handled by France fail to address the problem.
Another important point is that all the propositions so far and the Dublin system itself only cover asylum seekers. There is also an EU directive about asylum but individual countries still retain a lot of discretion regarding individual decisions and they sure as hell won't give it up. This leads to almost absurd disparities in the rate of successful asylum applications between different countries. I don't have recent statistics at hand but I think that between, say, Sweden and Greece, there is an order of magnitude of difference, for asylum seekers coming from the same country (e.g. maybe 70% of the people from Afghanistan who apply for asylum in Sweden are granted refugee status, compared to at most a few percents in Greece).
Because of this, even if the number of application was strictly proportional to the population, the outcome (the number of refugees, pictured on the NYT map) would still differ a lot. Also, the most pressing issue are the people arriving now but countries also differ in the way they treat people who have been refugees for a long time (IIRC, a few years ago Germany deemed Iraq a safe country and started deporting Iraqi citizens who were previously granted refugee status back to the country).
All this means that it will not be possible to have some European institution decide on asylum applications or spread refugees around. Any quota or burden sharing system will be just another way to spread applications very much like the Dublin system. But then all the issues mentioned above will remain mostly unsolved.
In this context, a country like Sweden really has two choices:
You cant's stop people from crossing the Mediterranean, you can't force Greece or Italy to put a lot of their own money in all this and take care of the problem for us and you can't force France to accept many asylum seekers to help everybody else. The only decision that is Sweden's to make is deport as many people as possible or accept to examine their applications.
Note that, if the Dublin system was working as intended, there is no particular reason why Sweden should receive more applications than other countries so the current system already protects it in theory. But fiddling with the rules used to assign applications to a country (basically forcing France or Poland to accept a few thousands applicants from elsewhere instead of sticking to the “country most responsible for entry” principle) will scarcely help in practice since those rules are widely ignored anyway.
Incidentally, it's only tangentially related to your question but it's important not to over interpret statistics on refugees and asylum seekers. They are very volatile and cover a range of different situations.
For example, the numbers for Germany skyrocketed with the Syrian crisis but were not particularly high before. Meanwhile, many applications in France and elsewhere do not come from people who fled Syria or Afghanistan but from people from relatively peaceful countries who lived in the country for some years before losing their status and only have this avenue to try to delay a removal.
