Almost certainly not.

The actual implementation and use of weregild is more complicated than Wikipedia, or even some basic history books covering the time, might have you believe. Money was not just a get out of, well not jail since it didn't exist yet, certainly some form of punishment card. Contrary to that chart you linked to, money was not just an excuse to straight up buy yourself out of negative consequences, and certainly could not have acted as such in practice. Now it is important to note that of course wealth did have certain impacts on the judicial system and wealthier, well connected, individuals were certainly less likely to face consequences for their actions.

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Historical context

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The idea behind the system of weregild was to pre-empt the cycle of feuding and to further centralize the King/Church (the distinction between the two is also not as strong as we might imagine) as the legitimate sources of authority in determining judicial outcomes, not feuding nobles who were liable to try and murder each other over their various disputes. This however was a rather difficult problem because Anglo-Saxon England, and certainly in the 700's, had very little in the way of effective government control. This system of weregild theoretically survived up until the Norman Conquest, who were more in favor of harsh physical punishment, including mutilations and executions for transgressions. Late Anglo-Saxon kings seem to have preferred letting the Church handle a lot of these issue, and they frequently preferred penitential action such as reconciliation, payment of weregild, forced pilgrimages, and various less...final.... physical punishments. (in certain cases)

The time frame you mention places us in the Heptarchy, the period of time when England was divided into seven kingdoms who were all nominally independent of each other and had shifting political allegiances to each other. A good way to envision it is essentially a giant game of "King of the Hill". Each polity/kingdom was trying to survive by subjugating all the others, but to get on top you needed help from the other kingdoms. Only once you were on top the other kingdoms that formerly supported you were all of a sudden less inclined to do so. There were periods of time when under particularly powerful rulers, who were declared Bretwaldas, or wide-rulers, were able to exercise some sort of authority beyond their own borders, but we should not assume that this made their word law in all corners of England.

During this time the kingdoms of the Heptarchy were starting to solidify into the power players of the Viking Age and the subsequent unification of England. Wessex, Mercia, and Northumbria were the heavy hitters by the end of this time period, while the kingdoms of East Anglia (which survived long enough to be overrun by the Vikings in the mid 9th century), Sussex, Essex, and Kent, were being edged out and increasingly subject to their larger neighbors, despite producing some of their own "wide rulers" in this time.

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Now with that out of the way, let's run through the scenario that you have laid out.

I assume that you're living in either Mercia or Northumbria, and the chart that is linked in Wikipedia does come from a Northumbrian law book, since those shared the closest borders to the Welsh polities and were more likely to have Welsh people of independent means within their borders and owning land. however it is worth noting that the Old English word, Wealas while used to refer to the native Britons enouh to become the word still used, is also rendered as foreigner more broadly.

But that's enough of that, lets delve into the meat of your question. Let's say you are a relatively prosperous freeman and you've recently come into some money. You've been feuding with your neighbor over the normal issues of land owners, whose property ends where, who gets the tribute/tax from one of the border village between your two estates, etc...

With this incessant conflict weighing upon you, you decided to use your windfall to pay off the weregild and murder the man. What happens next?

If you declare that you had been feuding with the man, and can bring witnesses to this feud as well as to your own character, and can afford to pay off the fine, this might be the end of the issue. The local magnate, such as a thane, earl, or even king depending on whose land you both live, might also ask for a cut of the payment so you may need to raise some additional funds and hope that this payment does indeed pre-empt any conflict.

Because lets be real, just paying your fine is not going to assuage the family of your victim, and if he indeed does have no kinsman in the area that can be reached (rather unlikely given the intensely familial dimension to politics, whoever he served under, such as an earl, might also not be pleased with your murder, payment or not. Regardless of your payment, there still may be an escalating feud between your two families that pulls in your other relations until you're in a small undeclared war. You may try to appeal to royal or Church authorities that you have done your duty by the letter of the law, but given that you're probably far from the centers of royal/Ecclesial authority, it might take a while for them to deal with your obviously very important issue. (To say nothing of any problems that they might need to attend to). Local Ecclesial officials might step in and demand some form of penance to be performed for murder regardless. Undergoing this would probably be a good idea as getting Ecclesial backing in a feud is a pretty powerful tool to undercut your opponent. You can expect to be abstaining from meat, wine, weapons, and so-on for a year or so, maybe more depending on how the priest or bishop interprets the situation.

However at this point your feud, despite the payment, could still spiral into a full on small war with communities taking sides based off of their ties to you or your victim's family. These could be through formalized patronage, marriage ties, economic ties, family ties etc....

This will eventually work its way out one way or another. The king may empower someone to sort this all out, probably a Church figure or another, more powerful, local lord. Or you might just kill each other enough that a halt to the fighting is necessitated.

However this is all assuming that your payment is even accepted. Given the absence of royal power in most of the country, the wronged family (or whoever is closely tied to them), may just prefer to extract their own justice at this time. After all the king is a long way off, and avenging a murder probably will stand up in court, even for a Welshman. Furthermore, if you've made this a habit, murdering wantonly and paying off your debts, you are not going to be winning hearts and minds of your neighbors or the other institutions that support you. The Church, royal figures, other land owners might not want to actually enforce your theoretical legal rights if you've got a reputation for murder, able to pay off or no.